Theorem bnj964 35240

Description: Technical lemma for bnj69 35307. 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.)

Hypotheses

Ref	Expression
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

Ref	Expression
bnj964	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜓″)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛   𝐷,𝑖   𝑖,𝐺   𝑅,𝑓,𝑖,𝑛   𝑖,𝑋   𝑓,𝑝,𝑖   𝑦,𝑓,𝑖,𝑛   𝑖,𝑚   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑚,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑚,𝑝)   𝐺(𝑦,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj964

Step	Hyp	Ref	Expression
1		nfv 1936	. . . 4 ⊢ Ⅎ𝑖(𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴)
2		bnj964.2	. . . . . . . 8 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3	2	bnj1095 35079	. . . . . . 7 ⊢ (𝜓 → ∀𝑖𝜓)
4		bnj964.3	. . . . . . 7 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5	3, 4	bnj1096 35080	. . . . . 6 ⊢ (𝜒 → ∀𝑖𝜒)
6	5	nf5i 2182	. . . . 5 ⊢ Ⅎ𝑖𝜒
7		nfv 1936	. . . . 5 ⊢ Ⅎ𝑖 𝑛 = suc 𝑚
8		nfv 1936	. . . . 5 ⊢ Ⅎ𝑖 𝑝 = suc 𝑛
9	6, 7, 8	nf3an 1923	. . . 4 ⊢ Ⅎ𝑖(𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)
10	1, 9	nfan 1921	. . 3 ⊢ Ⅎ𝑖((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
11		bnj255 35003	. . . . 5 ⊢ (((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)))
12		bnj645 35048	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → suc 𝑖 ∈ 𝑝)
13		simp3 1152	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑝 = suc 𝑛)
14	13	bnj706 35052	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → 𝑝 = suc 𝑛)
15		eleq2 2853	. . . . . . . . 9 ⊢ (𝑝 = suc 𝑛 → (suc 𝑖 ∈ 𝑝 ↔ suc 𝑖 ∈ suc 𝑛))
16	15	biimpac 482	. . . . . . . 8 ⊢ ((suc 𝑖 ∈ 𝑝 ∧ 𝑝 = suc 𝑛) → suc 𝑖 ∈ suc 𝑛)
17		elsuci 6417	. . . . . . . . 9 ⊢ (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖 ∈ 𝑛 ∨ suc 𝑖 = 𝑛))
18		eqcom 2771	. . . . . . . . . 10 ⊢ (suc 𝑖 = 𝑛 ↔ 𝑛 = suc 𝑖)
19	18	orbi2i 923	. . . . . . . . 9 ⊢ ((suc 𝑖 ∈ 𝑛 ∨ suc 𝑖 = 𝑛) ↔ (suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖))
20	17, 19	sylib 220	. . . . . . . 8 ⊢ (suc 𝑖 ∈ suc 𝑛 → (suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖))
21	16, 20	syl 17	. . . . . . 7 ⊢ ((suc 𝑖 ∈ 𝑝 ∧ 𝑝 = suc 𝑛) → (suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖))
22	12, 14, 21	syl2anc 593	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (suc 𝑖 ∈ 𝑛 ∨ 𝑛 = suc 𝑖))
23		df-3an 1101	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛) ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) ∧ suc 𝑖 ∈ 𝑛))
24	23	3anbi3i 1173	. . . . . . . . . . . 12 ⊢ (((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)))
25		bnj255 35003	. . . . . . . . . . . 12 ⊢ (((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)))
26	24, 25	bitr4i 280	. . . . . . . . . . 11 ⊢ (((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))
27		bnj345 35012	. . . . . . . . . . 11 ⊢ (((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)))
28		bnj252 35001	. . . . . . . . . . 11 ⊢ ((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))))
29	26, 27, 28	3bitri 299	. . . . . . . . . 10 ⊢ (((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))))
30	11	anbi2i 632	. . . . . . . . . 10 ⊢ ((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))))
31	29, 30	bitr4i 280	. . . . . . . . 9 ⊢ (((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)))
32		bnj964.96	. . . . . . . . 9 ⊢ (((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))
33	31, 32	sylbir 237	. . . . . . . 8 ⊢ ((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))
34	33	ex 416	. . . . . . 7 ⊢ (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)))
35		df-3an 1101	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖) ↔ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) ∧ 𝑛 = suc 𝑖))
36	35	3anbi3i 1173	. . . . . . . . . . . 12 ⊢ (((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)))
37		bnj255 35003	. . . . . . . . . . . 12 ⊢ (((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)))
38	36, 37	bitr4i 280	. . . . . . . . . . 11 ⊢ (((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))
39		bnj345 35012	. . . . . . . . . . 11 ⊢ (((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)))
40		bnj252 35001	. . . . . . . . . . 11 ⊢ ((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))))
41	38, 39, 40	3bitri 299	. . . . . . . . . 10 ⊢ (((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))))
42	11	anbi2i 632	. . . . . . . . . 10 ⊢ ((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))))
43	41, 42	bitr4i 280	. . . . . . . . 9 ⊢ (((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)))
44		bnj964.165	. . . . . . . . 9 ⊢ (((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))
45	43, 44	sylbir 237	. . . . . . . 8 ⊢ ((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))
46	45	ex 416	. . . . . . 7 ⊢ (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)))
47	34, 46	jaoi 868	. . . . . 6 ⊢ ((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)))
48	22, 47	mpcom 38	. . . . 5 ⊢ (((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))
49	11, 48	sylbir 237	. . . 4 ⊢ (((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	3expia 1135	. . 3 ⊢ (((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	10, 50	alrimi 2250	. 2 ⊢ (((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)))
52		bnj964.5	. . . . 5 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
53		vex 3460	. . . . 5 ⊢ 𝑝 ∈ V
54	2, 52, 53	bnj539 35188	. . . 4 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑝 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
55		bnj964.8	. . . 4 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
56		bnj964.12	. . . 4 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
57		bnj964.13	. . . 4 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
58	54, 55, 56, 57	bnj965 35239	. . 3 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑝 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
59	58	bnj115 35023	. 2 ⊢ (𝜓″ ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
60	51, 59	sylibr 236	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜓″)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099  ∀wal 1560   = wceq 1562   ∈ wcel 2144  ∀wral 3078  [wsbc 3746   ∪ cun 3904  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6350   Fn wfn 6518  ‘cfv 6523  ωcom 7848   ∧ w-bnj17 34984   predc-bnj14 34986   FrSe w-bnj15 34990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-suc 6354  df-iota 6479  df-fv 6531  df-bnj17 34985

This theorem is referenced by:  bnj910  35245