Theorem cvmscbv 32498

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 4838	. . . . . . 7 ⊢ (𝑠 = 𝑏 → ∪ 𝑠 = ∪ 𝑏)
3	2	eqeq1d 2821	. . . . . 6 ⊢ (𝑠 = 𝑏 → (∪ 𝑠 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑏 = (◡𝐹 “ 𝑘)))
4		ineq2 4181	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑑 → (𝑢 ∩ 𝑣) = (𝑢 ∩ 𝑑))
5	4	eqeq1d 2821	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑑 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝑢 ∩ 𝑑) = ∅))
6	5	cbvralvw 3448	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑑) = ∅)
7		sneq 4569	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑐 → {𝑢} = {𝑐})
8	7	difeq2d 4097	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → (𝑠 ∖ {𝑢}) = (𝑠 ∖ {𝑐}))
9		ineq1 4179	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑐 → (𝑢 ∩ 𝑑) = (𝑐 ∩ 𝑑))
10	9	eqeq1d 2821	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → ((𝑢 ∩ 𝑑) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅))
11	8, 10	raleqbidv 3400	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → (∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
12	6, 11	syl5bb 285	. . . . . . . . 9 ⊢ (𝑢 = 𝑐 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅))
13		reseq2 5841	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → (𝐹 ↾ 𝑢) = (𝐹 ↾ 𝑐))
14		oveq2 7156	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑐 → (𝐶 ↾t 𝑢) = (𝐶 ↾t 𝑐))
15	14	oveq1d 7163	. . . . . . . . . 10 ⊢ (𝑢 = 𝑐 → ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)))
16	13, 15	eleq12d 2905	. . . . . . . . 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 3448	. . . . . . 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 4090	. . . . . . . . . 10 ⊢ (𝑠 = 𝑏 → (𝑠 ∖ {𝑐}) = (𝑏 ∖ {𝑐}))
20	19	raleqdv 3414	. . . . . . . . 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 3402	. . . . . . 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	syl5bb 285	. . . . . 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 3490	. . . 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 5918	. . . . . . 7 ⊢ (𝑘 = 𝑎 → (◡𝐹 “ 𝑘) = (◡𝐹 “ 𝑎))
27	26	eqeq2d 2830	. . . . . 6 ⊢ (𝑘 = 𝑎 → (∪ 𝑏 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑏 = (◡𝐹 “ 𝑎)))
28		oveq2 7156	. . . . . . . . . 10 ⊢ (𝑘 = 𝑎 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑎))
29	28	oveq2d 7164	. . . . . . . . 9 ⊢ (𝑘 = 𝑎 → ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎)))
30	29	eleq2d 2896	. . . . . . . 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 3195	. . . . . 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 3479	. . . 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	syl5eq 2866	. . 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 5160	. 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 2842	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 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  {crab 3140   ∖ cdif 3931   ∩ cin 3933  ∅c0 4289  𝒫 cpw 4537  {csn 4559  ∪ cuni 4830   ↦ cmpt 5137  ^◡ccnv 5547   ↾ cres 5550   “ cima 5551  (class class class)co 7148   ↾_t crest 16686  Homeochmeo 22353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7151

This theorem is referenced by:  cvmsss2  32514  cvmliftmoi  32523  cvmlift  32539  cvmfo  32540  cvmlift3  32568