Theorem cvmscbv 35263

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 4918	. . . . . . 7 ⊢ (𝑠 = 𝑏 → ∪ 𝑠 = ∪ 𝑏)
3	2	eqeq1d 2739	. . . . . 6 ⊢ (𝑠 = 𝑏 → (∪ 𝑠 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑏 = (◡𝐹 “ 𝑘)))
4		ineq2 4214	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑑 → (𝑢 ∩ 𝑣) = (𝑢 ∩ 𝑑))
5	4	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑑 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑢 ∩ 𝑑) = ∅))
6	5	cbvralvw 3237	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑑) = ∅)
7		sneq 4636	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑐 → {𝑢} = {𝑐})
8	7	difeq2d 4126	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → (𝑠 ∖ {𝑢}) = (𝑠 ∖ {𝑐}))
9		ineq1 4213	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑐 → (𝑢 ∩ 𝑑) = (𝑐 ∩ 𝑑))
10	9	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → ((𝑢 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅))
11	8, 10	raleqbidv 3346	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → (∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
12	6, 11	bitrid 283	. . . . . . . . 9 ⊢ (𝑢 = 𝑐 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
13		reseq2 5992	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → (𝐹 ↾ 𝑢) = (𝐹 ↾ 𝑐))
14		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → (𝐶 ↾t 𝑢) = (𝐶 ↾t 𝑐))
15	14	oveq1d 7446	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))
16	13, 15	eleq12d 2835	. . . . . . . . 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 632	. . . . . . . 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 3237	. . . . . . 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 4119	. . . . . . . . . 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 631	. . . . . . . 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 3338	. . . . . . 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 283	. . . . . 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 632	. . . . 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 3447	. . . 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 6074	. . . . . . 7 ⊢ (𝑘 = 𝑎 → (◡𝐹 “ 𝑘) = (◡𝐹 “ 𝑎))
27	26	eqeq2d 2748	. . . . . 6 ⊢ (𝑘 = 𝑎 → (∪ 𝑏 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑏 = (◡𝐹 “ 𝑎)))
28		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑘 = 𝑎 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑎))
29	28	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑘 = 𝑎 → ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎)))
30	29	eleq2d 2827	. . . . . . . 8 ⊢ (𝑘 = 𝑎 → ((𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))
31	30	anbi2d 630	. . . . . . 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 3178	. . . . . 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 632	. . . . 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 3444	. . . 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 2789	. . 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 5255	. 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 2765	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))))})

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436   ∖ cdif 3948   ∩ cin 3950  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688  (class class class)co 7431   ↾_t crest 17465  Homeochmeo 23761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434

This theorem is referenced by:  cvmsss2  35279  cvmliftmoi  35288  cvmlift  35304  cvmfo  35305  cvmlift3  35333