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Theorem cvmscbv 30948
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 4410 . . . . . . 7 (𝑠 = 𝑏 𝑠 = 𝑏)
32eqeq1d 2623 . . . . . 6 (𝑠 = 𝑏 → ( 𝑠 = (𝐹𝑘) ↔ 𝑏 = (𝐹𝑘)))
4 ineq2 3786 . . . . . . . . . . . 12 (𝑣 = 𝑑 → (𝑢𝑣) = (𝑢𝑑))
54eqeq1d 2623 . . . . . . . . . . 11 (𝑣 = 𝑑 → ((𝑢𝑣) = ∅ ↔ (𝑢𝑑) = ∅))
65cbvralv 3159 . . . . . . . . . 10 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢𝑑) = ∅)
7 sneq 4158 . . . . . . . . . . . 12 (𝑢 = 𝑐 → {𝑢} = {𝑐})
87difeq2d 3706 . . . . . . . . . . 11 (𝑢 = 𝑐 → (𝑠 ∖ {𝑢}) = (𝑠 ∖ {𝑐}))
9 ineq1 3785 . . . . . . . . . . . 12 (𝑢 = 𝑐 → (𝑢𝑑) = (𝑐𝑑))
109eqeq1d 2623 . . . . . . . . . . 11 (𝑢 = 𝑐 → ((𝑢𝑑) = ∅ ↔ (𝑐𝑑) = ∅))
118, 10raleqbidv 3141 . . . . . . . . . 10 (𝑢 = 𝑐 → (∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅))
126, 11syl5bb 272 . . . . . . . . 9 (𝑢 = 𝑐 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅))
13 reseq2 5351 . . . . . . . . . 10 (𝑢 = 𝑐 → (𝐹𝑢) = (𝐹𝑐))
14 oveq2 6612 . . . . . . . . . . 11 (𝑢 = 𝑐 → (𝐶t 𝑢) = (𝐶t 𝑐))
1514oveq1d 6619 . . . . . . . . . 10 (𝑢 = 𝑐 → ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))
1613, 15eleq12d 2692 . . . . . . . . 9 (𝑢 = 𝑐 → ((𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))
1712, 16anbi12d 746 . . . . . . . 8 (𝑢 = 𝑐 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
1817cbvralv 3159 . . . . . . 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 3699 . . . . . . . . . 10 (𝑠 = 𝑏 → (𝑠 ∖ {𝑐}) = (𝑏 ∖ {𝑐}))
2019raleqdv 3133 . . . . . . . . 9 (𝑠 = 𝑏 → (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅))
2120anbi1d 740 . . . . . . . 8 (𝑠 = 𝑏 → ((∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
2221raleqbi1dv 3135 . . . . . . 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, 22syl5bb 272 . . . . . 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 746 . . . . 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 3185 . . . 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 5421 . . . . . . 7 (𝑘 = 𝑎 → (𝐹𝑘) = (𝐹𝑎))
2726eqeq2d 2631 . . . . . 6 (𝑘 = 𝑎 → ( 𝑏 = (𝐹𝑘) ↔ 𝑏 = (𝐹𝑎)))
28 oveq2 6612 . . . . . . . . . 10 (𝑘 = 𝑎 → (𝐽t 𝑘) = (𝐽t 𝑎))
2928oveq2d 6620 . . . . . . . . 9 (𝑘 = 𝑎 → ((𝐶t 𝑐)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))
3029eleq2d 2684 . . . . . . . 8 (𝑘 = 𝑎 → ((𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))
3130anbi2d 739 . . . . . . 7 (𝑘 = 𝑎 → ((∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))))
3231ralbidv 2980 . . . . . 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 746 . . . . 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 3177 . . . 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, 34syl5eq 2667 . . 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 4710 . 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 2643 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 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3552  cin 3554  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402  cmpt 4673  ccnv 5073  cres 5076  cima 5077  (class class class)co 6604  t crest 16002  Homeochmeo 21466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  cvmsss2  30964  cvmliftmoi  30973  cvmlift  30989  cvmfo  30990  cvmlift3  31018
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