Theorem cvmscbv 35771

Description: Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)

Hypothesis

Ref	Expression
iscvm.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmscbv	⊢ 𝑆 = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))})

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑘,𝑠,𝑢,𝑣   𝐶,𝑎,𝑏,𝑐,𝑘,𝑠,𝑢   𝐹,𝑎,𝑏,𝑐,𝑘,𝑠,𝑢   𝐽,𝑎,𝑏,𝑐,𝑘,𝑠,𝑢

Allowed substitution hints:   𝐶(𝑣, 𝑑)   𝑆(𝑣, 𝑢, 𝑘, 𝑠, 𝑎, 𝑏, 𝑐, 𝑑)   𝐹(𝑣, 𝑑)   𝐽(𝑣, 𝑑)

Proof of Theorem cvmscbv

Step	Hyp	Ref	Expression
1		iscvm.1	. 2 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
2		unieq 4888	. . . . . . 7 ⊢ (𝑠 = 𝑏 → ∪ 𝑠 = ∪ 𝑏)
3	2	eqeq1d 2768	. . . . . 6 ⊢ (𝑠 = 𝑏 → (∪ 𝑠 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑏 = (◡𝐹 “ 𝑘)))
4		ineq2 4170	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑑 → (𝑢 ∩ 𝑣) = (𝑢 ∩ 𝑑))
5	4	eqeq1d 2768	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑑 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑢 ∩ 𝑑) = ∅))
6	5	cbvralvw 3246	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑑) = ∅)
7		sneq 4604	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑐 → {𝑢} = {𝑐})
8	7	difeq2d 4084	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → (𝑠 ∖ {𝑢}) = (𝑠 ∖ {𝑐}))
9		ineq1 4169	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑐 → (𝑢 ∩ 𝑑) = (𝑐 ∩ 𝑑))
10	9	eqeq1d 2768	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → ((𝑢 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅))
11	8, 10	raleqbidv 3341	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → (∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
12	6, 11	bitrid 286	. . . . . . . . 9 ⊢ (𝑢 = 𝑐 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
13		reseq2 5978	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → (𝐹 ↾ 𝑢) = (𝐹 ↾ 𝑐))
14		oveq2 7431	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → (𝐶 ↾t 𝑢) = (𝐶 ↾t 𝑐))
15	14	oveq1d 7438	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))
16	13, 15	eleq12d 2860	. . . . . . . . 9 ⊢ (𝑢 = 𝑐 → ((𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))
17	12, 16	anbi12d 644	. . . . . . . 8 ⊢ (𝑢 = 𝑐 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))))
18	17	cbvralvw 3246	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))
19		difeq1 4077	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → (𝑠 ∖ {𝑐}) = (𝑏 ∖ {𝑐}))
20	19	raleqdv 3326	. . . . . . . . 9 ⊢ (𝑠 = 𝑏 → (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
21	20	anbi1d 643	. . . . . . . 8 ⊢ (𝑠 = 𝑏 → ((∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))))
22	21	raleqbi1dv 3336	. . . . . . 7 ⊢ (𝑠 = 𝑏 → (∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))) ↔ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))))
23	18, 22	bitrid 286	. . . . . 6 ⊢ (𝑠 = 𝑏 → (∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))))
24	3, 23	anbi12d 644	. . . . 5 ⊢ (𝑠 = 𝑏 → ((∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))) ↔ (∪ 𝑏 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))))
25	24	cbvrabv 3429	. . . 4 ⊢ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} = {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}
26		imaeq2 6063	. . . . . . 7 ⊢ (𝑘 = 𝑎 → (◡𝐹 “ 𝑘) = (◡𝐹 “ 𝑎))
27	26	eqeq2d 2777	. . . . . 6 ⊢ (𝑘 = 𝑎 → (∪ 𝑏 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑏 = (◡𝐹 “ 𝑎)))
28		oveq2 7431	. . . . . . . . . 10 ⊢ (𝑘 = 𝑎 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑎))
29	28	oveq2d 7439	. . . . . . . . 9 ⊢ (𝑘 = 𝑎 → ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎)))
30	29	eleq2d 2852	. . . . . . . 8 ⊢ (𝑘 = 𝑎 → ((𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))
31	30	anbi2d 642	. . . . . . 7 ⊢ (𝑘 = 𝑎 → ((∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎)))))
32	31	ralbidv 3191	. . . . . 6 ⊢ (𝑘 = 𝑎 → (∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))) ↔ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎)))))
33	27, 32	anbi12d 644	. . . . 5 ⊢ (𝑘 = 𝑎 → ((∪ 𝑏 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))) ↔ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))))
34	33	rabbidv 3426	. . . 4 ⊢ (𝑘 = 𝑎 → {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))} = {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))})
35	25, 34	eqtrid 2813	. . 3 ⊢ (𝑘 = 𝑎 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} = {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))})
36	35	cbvmptv 5220	. 2 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))})
37	1, 36	eqtri 2789	1 ⊢ 𝑆 = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))})

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419   ∖ cdif 3905   ∩ cin 3907  ∅c0 4289  𝒫 cpw 4567  {csn 4594  ∪ cuni 4877   ↦ cmpt 5197  ^◡ccnv 5665   ↾ cres 5668   “ cima 5669  (class class class)co 7423   ↾_t crest 17498  Homeochmeo 23947

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 2148  ax-9 2156  ax-ext 2738

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-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fv 6551  df-ov 7426

This theorem is used by:  cvmsss2  35787  cvmliftmoi  35796  cvmlift  35812  cvmfo  35813  cvmlift3  35841