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Metamath Proof Explorer

Theorem frr3g

Description: Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 . (Contributed by Scott Fenton, 10-Feb-2011) (Revised by Mario Carneiro, 26-Jun-2015)

Ref Expression
Assertion frr3g ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐹 = 𝐺 )

Proof

Step Hyp Ref Expression
1 ra4v ⊢ ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
2 r19.26 ⊢ ( ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) )
3 2 anbi2i ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) ↔ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) )
4 fveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) )
5 id ⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 )
6 predeq3 ⊢ ( 𝑦 = 𝑧 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑧 ) )
7 6 reseq2d ⊢ ( 𝑦 = 𝑧 → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
8 5 7 oveq12d ⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
9 4 8 eqeq12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) )
10 fveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) )
11 6 reseq2d ⊢ ( 𝑦 = 𝑧 → ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
12 5 11 oveq12d ⊢ ( 𝑦 = 𝑧 → ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
13 10 12 eqeq12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ↔ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) )
14 9 13 anbi12d ⊢ ( 𝑦 = 𝑧 → ( ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) ) )
15 14 rspcva ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) )
16 eqtr3 ⊢ ( ( ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝐹 ‘ 𝑧 ) )
17 16 eqcomd ⊢ ( ( ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
18 eqtr3 ⊢ ( ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) )
19 18 ex ⊢ ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) → ( ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
20 17 19 syl ⊢ ( ( ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
21 20 expimpd ⊢ ( ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) → ( ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
22 predss ⊢ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴
23 fvreseq ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ 𝐴 ) → ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ↔ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
24 22 23 mpan2 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ↔ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
25 24 biimpar ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) = ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) )
26 25 oveq2d ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
27 26 eqcomd ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) )
28 21 27 syl11 ⊢ ( ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
29 28 expd ⊢ ( ( ( 𝐹 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ∧ ( 𝐺 ‘ 𝑧 ) = ( 𝑧 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ) → ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
30 15 29 syl ⊢ ( ( 𝑧 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
31 30 ex ⊢ ( 𝑧 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) → ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) ) )
32 31 com23 ⊢ ( 𝑧 ∈ 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) ) )
33 32 impd ⊢ ( 𝑧 ∈ 𝐴 → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
34 3 33 biimtrrid ⊢ ( 𝑧 ∈ 𝐴 → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
35 34 a2d ⊢ ( 𝑧 ∈ 𝐴 → ( ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
36 1 35 syl5 ⊢ ( 𝑧 ∈ 𝐴 → ( ∀ 𝑤 ∈ Pred ( 𝑅 , 𝐴 , 𝑧 ) ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
37 fveq2 ⊢ ( 𝑧 = 𝑤 → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑤 ) )
38 fveq2 ⊢ ( 𝑧 = 𝑤 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑤 ) )
39 37 38 eqeq12d ⊢ ( 𝑧 = 𝑤 → ( ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ↔ ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) )
40 39 imbi2d ⊢ ( 𝑧 = 𝑤 → ( ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ↔ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) ) )
41 36 40 frins2 ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑧 ∈ 𝐴 ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
42 rsp ⊢ ( ∀ 𝑧 ∈ 𝐴 ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) → ( 𝑧 ∈ 𝐴 → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
43 41 42 syl ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑧 ∈ 𝐴 → ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
44 43 com3r ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑧 ∈ 𝐴 → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
45 44 an4s ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑧 ∈ 𝐴 → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
46 45 com12 ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝑧 ∈ 𝐴 → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
47 46 3impib ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝑧 ∈ 𝐴 → ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
48 47 ralrimiv ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) )
49 eqid ⊢ 𝐴 = 𝐴
50 48 49 jctil ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐴 = 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) )
51 eqfnfv2 ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐹 = 𝐺 ↔ ( 𝐴 = 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
52 51 ad2ant2r ⊢ ( ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 = 𝐺 ↔ ( 𝐴 = 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
53 52 3adant1 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → ( 𝐹 = 𝐺 ↔ ( 𝐴 = 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ( 𝐹 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) ) ) )
54 50 53 mpbird ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ∧ ( 𝐺 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝐺 ‘ 𝑦 ) = ( 𝑦 𝐻 ( 𝐺 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) ) → 𝐹 = 𝐺 )