Theorem bnj966 35494

Description: Technical lemma for bnj69 35560. 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
bnj966.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj966.10	⊢ 𝐷 = (ω ∖ {∅})
bnj966.12	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj966.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj966.44	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
bnj966.53	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)

Assertion

Ref	Expression
bnj966	⊢ (((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))

Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑚   𝑦,𝑛

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

Proof of Theorem bnj966

Step	Hyp	Ref	Expression
1		bnj966.53	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐺 Fn 𝑝)
2	1	fnfund 6629	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → Fun 𝐺)
3	2	3adant3 1150	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → Fun 𝐺)
4		opex 5432	. . . . . . 7 ⊢ ⟨𝑛, 𝐶⟩ ∈ V
5	4	snid 4623	. . . . . 6 ⊢ ⟨𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩}
6		elun2 4129	. . . . . 6 ⊢ (⟨𝑛, 𝐶⟩ ∈ {⟨𝑛, 𝐶⟩} → ⟨𝑛, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑛, 𝐶⟩}))
7	5, 6	ax-mp 5	. . . . 5 ⊢ ⟨𝑛, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑛, 𝐶⟩})
8		bnj966.13	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
9	7, 8	eleqtrri 2859	. . . 4 ⊢ ⟨𝑛, 𝐶⟩ ∈ 𝐺
10		funopfv 6923	. . . 4 ⊢ (Fun 𝐺 → (⟨𝑛, 𝐶⟩ ∈ 𝐺 → (𝐺‘𝑛) = 𝐶))
11	3, 9, 10	mpisyl 22	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘𝑛) = 𝐶)
12		simp22 1226	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑛 = suc 𝑚)
13		simp33 1230	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑛 = suc 𝑖)
14		bnj551 35293	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑛 = suc 𝑖) → 𝑚 = 𝑖)
15	12, 13, 14	syl2anc 596	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑚 = 𝑖)
16		suceq 6421	. . . . . . . 8 ⊢ (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖)
17	16	eqeq2d 2771	. . . . . . 7 ⊢ (𝑚 = 𝑖 → (𝑛 = suc 𝑚 ↔ 𝑛 = suc 𝑖))
18	17	biimpac 484	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → 𝑛 = suc 𝑖)
19	18	fveq2d 6878	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → (𝐺‘𝑛) = (𝐺‘suc 𝑖))
20		bnj966.12	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
21		fveq2 6874	. . . . . . . 8 ⊢ (𝑚 = 𝑖 → (𝑓‘𝑚) = (𝑓‘𝑖))
22	21	bnj1113 35336	. . . . . . 7 ⊢ (𝑚 = 𝑖 → ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
23	20, 22	eqtrid 2807	. . . . . 6 ⊢ (𝑚 = 𝑖 → 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
24	23	adantl 487	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
25	19, 24	eqeq12d 2776	. . . 4 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑚 = 𝑖) → ((𝐺‘𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
26	12, 15, 25	syl2anc 596	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → ((𝐺‘𝑛) = 𝐶 ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
27	11, 26	mpbid 235	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
28		bnj966.44	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
29	28	3adant3 1150	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝐶 ∈ V)
30		bnj966.3	. . . . . . . 8 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
31	30	bnj1235 35354	. . . . . . 7 ⊢ (𝜒 → 𝑓 Fn 𝑛)
32	31	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
33	32	3ad2ant2 1152	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑓 Fn 𝑛)
34		simp23 1227	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑝 = suc 𝑛)
35	29, 33, 34, 13	bnj951 35326	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖))
36		bnj966.10	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
37	36	bnj923 35319	. . . . . . . 8 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
38	30, 37	bnj769 35313	. . . . . . 7 ⊢ (𝜒 → 𝑛 ∈ ω)
39	38	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ ω)
40		simp3 1156	. . . . . 6 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖) → 𝑛 = suc 𝑖)
41	39, 40	bnj240 35250	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝑛 ∈ ω ∧ 𝑛 = suc 𝑖))
42		vex 3454	. . . . . . 7 ⊢ 𝑖 ∈ V
43	42	bnj216 35283	. . . . . 6 ⊢ (𝑛 = suc 𝑖 → 𝑖 ∈ 𝑛)
44	43	adantl 487	. . . . 5 ⊢ ((𝑛 ∈ ω ∧ 𝑛 = suc 𝑖) → 𝑖 ∈ 𝑛)
45	41, 44	syl 18	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → 𝑖 ∈ 𝑛)
46		bnj658 35302	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛))
47	46	anim1i 627	. . . . . 6 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) ∧ 𝑖 ∈ 𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ 𝑛))
48		df-bnj17 35238	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ 𝑛))
49	47, 48	sylibr 237	. . . . 5 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) ∧ 𝑖 ∈ 𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛))
50	8	bnj945 35324	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
51	49, 50	syl 18	. . . 4 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑛 = suc 𝑖) ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
52	35, 45, 51	syl2anc 596	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑛 = suc 𝑖)) → (𝐺‘𝑖) = (𝑓‘𝑖))
53	20, 8	bnj958 35490	. . . . 5 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))
54	53	bnj956 35327	. . . 4 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
55	54	eqeq2d 2771	. . 3 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ((𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
56	52, 55	syl 18	. 2 ⊢ (((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) ↔ (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
57	27, 56	mpbird 260	1 ⊢ (((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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897  ∅c0 4279  {csn 4584  ⟨cop 4590  ∪ ciun 4951  suc csuc 6354  Fun wfun 6522   Fn wfn 6523  ‘cfv 6528  ωcom 7861   ∧ w-bnj17 35237   predc-bnj14 35239   FrSe w-bnj15 35243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-id 5543  df-eprel 5548  df-fr 5601  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-res 5660  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536  df-bnj17 35238

This theorem is used by:  bnj910  35498