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Metamath Proof Explorer

Theorem frr1

Description: Law of general well-founded recursion, part one. This theorem and the following two drop the partial order requirement from fpr1 , fpr2 , and fpr3 , which requires using the axiom of infinity (Contributed by Scott Fenton, 11-Sep-2023)

Ref Expression
Hypothesis frr.1 ⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion frr1 ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )

Proof

Step Hyp Ref Expression
1 frr.1 ⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 )
2 eqid ⊢ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) }
3 2 frrlem1 ⊢ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
4 3 1 frrlem15 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑔 ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ∧ ℎ ∈ { 𝑎 ∣ ∃ 𝑏 ( 𝑎 Fn 𝑏 ∧ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑐 ∈ 𝑏 Pred ( 𝑅 , 𝐴 , 𝑐 ) ⊆ 𝑏 ) ∧ ∀ 𝑐 ∈ 𝑏 ( 𝑎 ‘ 𝑐 ) = ( 𝑐 𝐺 ( 𝑎 ↾ Pred ( 𝑅 , 𝐴 , 𝑐 ) ) ) ) } ) ) → ( ( 𝑥 𝑔 𝑢 ∧ 𝑥 ℎ 𝑣 ) → 𝑢 = 𝑣 ) )
5 3 1 4 frrlem9 ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → Fun 𝐹 )
6 eqid ⊢ ( ( 𝐹 ↾ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } ) = ( ( 𝐹 ↾ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ) ∪ { ⟨ 𝑧 , ( 𝑧 𝐺 ( 𝐹 ↾ Pred ( 𝑅 , 𝐴 , 𝑧 ) ) ) ⟩ } )
7 simpl ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
8 predres ⊢ Pred ( 𝑅 , 𝐴 , 𝑧 ) = Pred ( ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 )
9 relres ⊢ Rel ( 𝑅 ↾ 𝐴 )
10 ssttrcl ⊢ ( Rel ( 𝑅 ↾ 𝐴 ) → ( 𝑅 ↾ 𝐴 ) ⊆ t++ ( 𝑅 ↾ 𝐴 ) )
11 predrelss ⊢ ( ( 𝑅 ↾ 𝐴 ) ⊆ t++ ( 𝑅 ↾ 𝐴 ) → Pred ( ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ⊆ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) )
12 9 10 11 mp2b ⊢ Pred ( ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ⊆ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 )
13 8 12 eqsstri ⊢ Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 )
14 13 a1i ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑧 ) ⊆ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) )
15 frrlem16 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → ∀ 𝑎 ∈ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) Pred ( 𝑅 , 𝐴 , 𝑎 ) ⊆ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) )
16 ttrclse ⊢ ( 𝑅 Se 𝐴 → t++ ( 𝑅 ↾ 𝐴 ) Se 𝐴 )
17 setlikespec ⊢ ( ( 𝑧 ∈ 𝐴 ∧ t++ ( 𝑅 ↾ 𝐴 ) Se 𝐴 ) → Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ∈ V )
18 17 ancoms ⊢ ( ( t++ ( 𝑅 ↾ 𝐴 ) Se 𝐴 ∧ 𝑧 ∈ 𝐴 ) → Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ∈ V )
19 16 18 sylan ⊢ ( ( 𝑅 Se 𝐴 ∧ 𝑧 ∈ 𝐴 ) → Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ∈ V )
20 19 adantll ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ∈ V )
21 predss ⊢ Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ⊆ 𝐴
22 21 a1i ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → Pred ( t++ ( 𝑅 ↾ 𝐴 ) , 𝐴 , 𝑧 ) ⊆ 𝐴 )
23 difss ⊢ ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴
24 frmin ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ( 𝐴 ∖ dom 𝐹 ) ⊆ 𝐴 ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) ) → ∃ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ )
25 23 24 mpanr1 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐴 ∖ dom 𝐹 ) ≠ ∅ ) → ∃ 𝑧 ∈ ( 𝐴 ∖ dom 𝐹 ) Pred ( 𝑅 , ( 𝐴 ∖ dom 𝐹 ) , 𝑧 ) = ∅ )
26 3 1 4 6 7 14 15 20 22 25 frrlem14 ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → dom 𝐹 = 𝐴 )
27 df-fn ⊢ ( 𝐹 Fn 𝐴 ↔ ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) )
28 5 26 27 sylanbrc ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn 𝐴 )