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Metamath Proof Explorer

Theorem bnj535

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj535.1 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
bnj535.2 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj535.3 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
bnj535.4 ⊢ ( 𝜏 ↔ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) )
Assertion bnj535 ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) → 𝐺 Fn 𝑛 )

Proof

Step Hyp Ref Expression
1 bnj535.1 ⊢ ( 𝜑′ ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )
2 bnj535.2 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑚 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3 bnj535.3 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
4 bnj535.4 ⊢ ( 𝜏 ↔ ( 𝜑′ ∧ 𝜓′ ∧ 𝑚 ∈ ω ∧ 𝑝 ∈ 𝑚 ) )
5 bnj422 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) ↔ ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏 ) )
6 bnj251 ⊢ ( ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ∧ 𝑅 FrSe 𝐴 ∧ 𝜏 ) ↔ ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) ) ) )
7 5 6 bitri ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) ↔ ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) ) ) )
8 fvex ⊢ ( 𝑓 ‘ 𝑝 ) ∈ V
9 1 2 4 bnj518 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) → ∀ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )
10 iunexg ⊢ ( ( ( 𝑓 ‘ 𝑝 ) ∈ V ∧ ∀ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )
11 8 9 10 sylancr ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V )
12 vex ⊢ 𝑚 ∈ V
13 12 bnj519 ⊢ ( ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V → Fun { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
14 11 13 syl ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) → Fun { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } )
15 dmsnopg ⊢ ( ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ∈ V → dom { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } = { 𝑚 } )
16 11 15 syl ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) → dom { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } = { 𝑚 } )
17 14 16 bnj1422 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) → { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } Fn { 𝑚 } )
18 disjcsn ⊢ ( 𝑚 ∩ { 𝑚 } ) = ∅
19 fnun ⊢ ( ( ( 𝑓 Fn 𝑚 ∧ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } Fn { 𝑚 } ) ∧ ( 𝑚 ∩ { 𝑚 } ) = ∅ ) → ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } ) Fn ( 𝑚 ∪ { 𝑚 } ) )
20 18 19 mpan2 ⊢ ( ( 𝑓 Fn 𝑚 ∧ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } Fn { 𝑚 } ) → ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } ) Fn ( 𝑚 ∪ { 𝑚 } ) )
21 17 20 sylan2 ⊢ ( ( 𝑓 Fn 𝑚 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) ) → ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } ) Fn ( 𝑚 ∪ { 𝑚 } ) )
22 3 fneq1i ⊢ ( 𝐺 Fn ( 𝑚 ∪ { 𝑚 } ) ↔ ( 𝑓 ∪ { ⟨ 𝑚 , ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑝 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ⟩ } ) Fn ( 𝑚 ∪ { 𝑚 } ) )
23 21 22 sylibr ⊢ ( ( 𝑓 Fn 𝑚 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) ) → 𝐺 Fn ( 𝑚 ∪ { 𝑚 } ) )
24 fneq2 ⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) → ( 𝐺 Fn 𝑛 ↔ 𝐺 Fn ( 𝑚 ∪ { 𝑚 } ) ) )
25 23 24 imbitrrid ⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) → ( ( 𝑓 Fn 𝑚 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) ) → 𝐺 Fn 𝑛 ) )
26 25 imp ⊢ ( ( 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ ( 𝑓 Fn 𝑚 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝜏 ) ) ) → 𝐺 Fn 𝑛 )
27 7 26 sylbi ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝑛 = ( 𝑚 ∪ { 𝑚 } ) ∧ 𝑓 Fn 𝑚 ) → 𝐺 Fn 𝑛 )