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Theorem cvmscbv 34901
Description: Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
Hypothesis
Ref Expression
iscvm.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmscbv 𝑆 = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 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
StepHypRef Expression
1 iscvm.1 . 2 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
2 unieq 4923 . . . . . . 7 (𝑠 = 𝑏 𝑠 = 𝑏)
32eqeq1d 2730 . . . . . 6 (𝑠 = 𝑏 → ( 𝑠 = (𝐹𝑘) ↔ 𝑏 = (𝐹𝑘)))
4 ineq2 4208 . . . . . . . . . . . 12 (𝑣 = 𝑑 → (𝑢𝑣) = (𝑢𝑑))
54eqeq1d 2730 . . . . . . . . . . 11 (𝑣 = 𝑑 → ((𝑢𝑣) = ∅ ↔ (𝑢𝑑) = ∅))
65cbvralvw 3232 . . . . . . . . . 10 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢𝑑) = ∅)
7 sneq 4642 . . . . . . . . . . . 12 (𝑢 = 𝑐 → {𝑢} = {𝑐})
87difeq2d 4122 . . . . . . . . . . 11 (𝑢 = 𝑐 → (𝑠 ∖ {𝑢}) = (𝑠 ∖ {𝑐}))
9 ineq1 4207 . . . . . . . . . . . 12 (𝑢 = 𝑐 → (𝑢𝑑) = (𝑐𝑑))
109eqeq1d 2730 . . . . . . . . . . 11 (𝑢 = 𝑐 → ((𝑢𝑑) = ∅ ↔ (𝑐𝑑) = ∅))
118, 10raleqbidv 3340 . . . . . . . . . 10 (𝑢 = 𝑐 → (∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅))
126, 11bitrid 282 . . . . . . . . 9 (𝑢 = 𝑐 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅))
13 reseq2 5984 . . . . . . . . . 10 (𝑢 = 𝑐 → (𝐹𝑢) = (𝐹𝑐))
14 oveq2 7434 . . . . . . . . . . 11 (𝑢 = 𝑐 → (𝐶t 𝑢) = (𝐶t 𝑐))
1514oveq1d 7441 . . . . . . . . . 10 (𝑢 = 𝑐 → ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))
1613, 15eleq12d 2823 . . . . . . . . 9 (𝑢 = 𝑐 → ((𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))
1712, 16anbi12d 630 . . . . . . . 8 (𝑢 = 𝑐 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
1817cbvralvw 3232 . . . . . . 7 (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))
19 difeq1 4115 . . . . . . . . . 10 (𝑠 = 𝑏 → (𝑠 ∖ {𝑐}) = (𝑏 ∖ {𝑐}))
2019raleqdv 3323 . . . . . . . . 9 (𝑠 = 𝑏 → (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅))
2120anbi1d 629 . . . . . . . 8 (𝑠 = 𝑏 → ((∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
2221raleqbi1dv 3331 . . . . . . 7 (𝑠 = 𝑏 → (∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
2318, 22bitrid 282 . . . . . 6 (𝑠 = 𝑏 → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
243, 23anbi12d 630 . . . . 5 (𝑠 = 𝑏 → (( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))) ↔ ( 𝑏 = (𝐹𝑘) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))))
2524cbvrabv 3441 . . . 4 {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} = {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑘) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))}
26 imaeq2 6064 . . . . . . 7 (𝑘 = 𝑎 → (𝐹𝑘) = (𝐹𝑎))
2726eqeq2d 2739 . . . . . 6 (𝑘 = 𝑎 → ( 𝑏 = (𝐹𝑘) ↔ 𝑏 = (𝐹𝑎)))
28 oveq2 7434 . . . . . . . . . 10 (𝑘 = 𝑎 → (𝐽t 𝑘) = (𝐽t 𝑎))
2928oveq2d 7442 . . . . . . . . 9 (𝑘 = 𝑎 → ((𝐶t 𝑐)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))
3029eleq2d 2815 . . . . . . . 8 (𝑘 = 𝑎 → ((𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))
3130anbi2d 628 . . . . . . 7 (𝑘 = 𝑎 → ((∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))))
3231ralbidv 3175 . . . . . 6 (𝑘 = 𝑎 → (∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))))
3327, 32anbi12d 630 . . . . 5 (𝑘 = 𝑎 → (( 𝑏 = (𝐹𝑘) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))) ↔ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))))
3433rabbidv 3438 . . . 4 (𝑘 = 𝑎 → {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑘) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))} = {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))})
3525, 34eqtrid 2780 . . 3 (𝑘 = 𝑎 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} = {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))})
3635cbvmptv 5265 . 2 (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))}) = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))})
371, 36eqtri 2756 1 𝑆 = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))})
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  wral 3058  {crab 3430  cdif 3946  cin 3948  c0 4326  𝒫 cpw 4606  {csn 4632   cuni 4912  cmpt 5235  ccnv 5681  cres 5684  cima 5685  (class class class)co 7426  t crest 17409  Homeochmeo 23677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  cvmsss2  34917  cvmliftmoi  34926  cvmlift  34942  cvmfo  34943  cvmlift3  34971
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