Metamath Proof Explorer

Theorem bnj964

Description: Technical lemma for bnj69 . 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 bnj964.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
bnj964.3 ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
bnj964.5 ⊢ ( 𝜓′ ↔ [ 𝑝 / 𝑛 ] 𝜓 )
bnj964.8 ⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓′ )
bnj964.12 ⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
bnj964.13 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
bnj964.96 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
bnj964.165 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
Assertion bnj964 ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝜓″ )

Proof

Step Hyp Ref Expression
1 bnj964.2 ⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2 bnj964.3 ⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
3 bnj964.5 ⊢ ( 𝜓′ ↔ [ 𝑝 / 𝑛 ] 𝜓 )
4 bnj964.8 ⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓′ )
5 bnj964.12 ⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
6 bnj964.13 ⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
7 bnj964.96 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
8 bnj964.165 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
9 nfv ⊢ Ⅎ 𝑖 ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 )
10 1 bnj1095 ⊢ ( 𝜓 → ∀ 𝑖 𝜓 )
11 10 2 bnj1096 ⊢ ( 𝜒 → ∀ 𝑖 𝜒 )
12 11 nf5i ⊢ Ⅎ 𝑖 𝜒
13 nfv ⊢ Ⅎ 𝑖 𝑛 = suc 𝑚
14 nfv ⊢ Ⅎ 𝑖 𝑝 = suc 𝑛
15 12 13 14 nf3an ⊢ Ⅎ 𝑖 ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 )
16 9 15 nfan ⊢ Ⅎ 𝑖 ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
17 bnj255 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) )
18 bnj645 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → suc 𝑖 ∈ 𝑝 )
19 simp3 ⊢ ( ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) → 𝑝 = suc 𝑛 )
20 19 bnj706 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → 𝑝 = suc 𝑛 )
21 eleq2 ⊢ ( 𝑝 = suc 𝑛 → ( suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛 ) )
22 21 biimpac ⊢ ( ( suc 𝑖 ∈ 𝑝 ∧ 𝑝 = suc 𝑛 ) → suc 𝑖 ∈ suc 𝑛 )
23 elsuci ⊢ ( suc 𝑖 ∈ suc 𝑛 → ( suc 𝑖 ∈ 𝑛 ∨ suc 𝑖 = 𝑛 ) )
24 eqcom ⊢ ( suc 𝑖 = 𝑛 ↔ 𝑛 = suc 𝑖 )
25 24 orbi2i ⊢ ( ( suc 𝑖 ∈ 𝑛 ∨ suc 𝑖 = 𝑛 ) ↔ ( suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖 ) )
26 23 25 sylib ⊢ ( suc 𝑖 ∈ suc 𝑛 → ( suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖 ) )
27 22 26 syl ⊢ ( ( suc 𝑖 ∈ 𝑝 ∧ 𝑝 = suc 𝑛 ) → ( suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖 ) )
28 18 20 27 syl2anc ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖 ) )
29 df-3an ⊢ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ↔ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ suc 𝑖 ∈ 𝑛 ) )
30 29 3anbi3i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ suc 𝑖 ∈ 𝑛 ) ) )
31 bnj255 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ suc 𝑖 ∈ 𝑛 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ suc 𝑖 ∈ 𝑛 ) ) )
32 30 31 bitr4i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ suc 𝑖 ∈ 𝑛 ) )
33 bnj345 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ suc 𝑖 ∈ 𝑛 ) ↔ ( suc 𝑖 ∈ 𝑛 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) )
34 bnj252 ⊢ ( ( suc 𝑖 ∈ 𝑛 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ↔ ( suc 𝑖 ∈ 𝑛 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ) )
35 32 33 34 3bitri ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ) ↔ ( suc 𝑖 ∈ 𝑛 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ) )
36 17 anbi2i ⊢ ( ( suc 𝑖 ∈ 𝑛 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ↔ ( suc 𝑖 ∈ 𝑛 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ) )
37 35 36 bitr4i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛 ) ) ↔ ( suc 𝑖 ∈ 𝑛 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) )
38 37 7 sylbir ⊢ ( ( suc 𝑖 ∈ 𝑛 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
39 38 ex ⊢ ( suc 𝑖 ∈ 𝑛 → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
40 df-3an ⊢ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ↔ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ 𝑛 = suc 𝑖 ) )
41 40 3anbi3i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ 𝑛 = suc 𝑖 ) ) )
42 bnj255 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ 𝑛 = suc 𝑖 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ 𝑛 = suc 𝑖 ) ) )
43 41 42 bitr4i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ 𝑛 = suc 𝑖 ) )
44 bnj345 ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ∧ 𝑛 = suc 𝑖 ) ↔ ( 𝑛 = suc 𝑖 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) )
45 bnj252 ⊢ ( ( 𝑛 = suc 𝑖 ∧ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ↔ ( 𝑛 = suc 𝑖 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ) )
46 43 44 45 3bitri ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ) ↔ ( 𝑛 = suc 𝑖 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ) )
47 17 anbi2i ⊢ ( ( 𝑛 = suc 𝑖 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ↔ ( 𝑛 = suc 𝑖 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) ) )
48 46 47 bitr4i ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖 ) ) ↔ ( 𝑛 = suc 𝑖 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) )
49 48 8 sylbir ⊢ ( ( 𝑛 = suc 𝑖 ∧ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
50 49 ex ⊢ ( 𝑛 = suc 𝑖 → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
51 39 50 jaoi ⊢ ( ( suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖 ) → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
52 28 51 mpcom ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
53 17 52 sylbir ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ∧ ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
54 53 3expia ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
55 16 54 alrimi ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → ∀ 𝑖 ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
56 vex ⊢ 𝑝 ∈ V
57 1 3 56 bnj539 ⊢ ( 𝜓′ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑝 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
58 57 4 5 6 bnj965 ⊢ ( 𝜓″ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑝 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
59 58 bnj115 ⊢ ( 𝜓″ ↔ ∀ 𝑖 ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
60 55 59 sylibr ⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝜓″ )