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Theorem cvmscbv 32633
 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 4812 . . . . . . 7 (𝑠 = 𝑏 𝑠 = 𝑏)
32eqeq1d 2800 . . . . . 6 (𝑠 = 𝑏 → ( 𝑠 = (𝐹𝑘) ↔ 𝑏 = (𝐹𝑘)))
4 ineq2 4133 . . . . . . . . . . . 12 (𝑣 = 𝑑 → (𝑢𝑣) = (𝑢𝑑))
54eqeq1d 2800 . . . . . . . . . . 11 (𝑣 = 𝑑 → ((𝑢𝑣) = ∅ ↔ (𝑢𝑑) = ∅))
65cbvralvw 3396 . . . . . . . . . 10 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢𝑑) = ∅)
7 sneq 4535 . . . . . . . . . . . 12 (𝑢 = 𝑐 → {𝑢} = {𝑐})
87difeq2d 4050 . . . . . . . . . . 11 (𝑢 = 𝑐 → (𝑠 ∖ {𝑢}) = (𝑠 ∖ {𝑐}))
9 ineq1 4131 . . . . . . . . . . . 12 (𝑢 = 𝑐 → (𝑢𝑑) = (𝑐𝑑))
109eqeq1d 2800 . . . . . . . . . . 11 (𝑢 = 𝑐 → ((𝑢𝑑) = ∅ ↔ (𝑐𝑑) = ∅))
118, 10raleqbidv 3354 . . . . . . . . . 10 (𝑢 = 𝑐 → (∀𝑑 ∈ (𝑠 ∖ {𝑢})(𝑢𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅))
126, 11syl5bb 286 . . . . . . . . 9 (𝑢 = 𝑐 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅))
13 reseq2 5814 . . . . . . . . . 10 (𝑢 = 𝑐 → (𝐹𝑢) = (𝐹𝑐))
14 oveq2 7144 . . . . . . . . . . 11 (𝑢 = 𝑐 → (𝐶t 𝑢) = (𝐶t 𝑐))
1514oveq1d 7151 . . . . . . . . . 10 (𝑢 = 𝑐 → ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))
1613, 15eleq12d 2884 . . . . . . . . 9 (𝑢 = 𝑐 → ((𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))
1712, 16anbi12d 633 . . . . . . . 8 (𝑢 = 𝑐 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
1817cbvralvw 3396 . . . . . . 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 4043 . . . . . . . . . 10 (𝑠 = 𝑏 → (𝑠 ∖ {𝑐}) = (𝑏 ∖ {𝑐}))
2019raleqdv 3364 . . . . . . . . 9 (𝑠 = 𝑏 → (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ↔ ∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅))
2120anbi1d 632 . . . . . . . 8 (𝑠 = 𝑏 → ((∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)))))
2221raleqbi1dv 3356 . . . . . . 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 286 . . . . . 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 633 . . . . 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 3439 . . . 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 5893 . . . . . . 7 (𝑘 = 𝑎 → (𝐹𝑘) = (𝐹𝑎))
2726eqeq2d 2809 . . . . . 6 (𝑘 = 𝑎 → ( 𝑏 = (𝐹𝑘) ↔ 𝑏 = (𝐹𝑎)))
28 oveq2 7144 . . . . . . . . . 10 (𝑘 = 𝑎 → (𝐽t 𝑘) = (𝐽t 𝑎))
2928oveq2d 7152 . . . . . . . . 9 (𝑘 = 𝑎 → ((𝐶t 𝑐)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))
3029eleq2d 2875 . . . . . . . 8 (𝑘 = 𝑎 → ((𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))
3130anbi2d 631 . . . . . . 7 (𝑘 = 𝑎 → ((∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))) ↔ (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎)))))
3231ralbidv 3162 . . . . . 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 633 . . . . 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 3427 . . . 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 2845 . . 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 5134 . 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 2821 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ∖ cdif 3878   ∩ cin 3880  ∅c0 4243  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   ↦ cmpt 5111  ◡ccnv 5519   ↾ cres 5522   “ cima 5523  (class class class)co 7136   ↾t crest 16689  Homeochmeo 22368 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-xp 5526  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fv 6333  df-ov 7139 This theorem is referenced by:  cvmsss2  32649  cvmliftmoi  32658  cvmlift  32674  cvmfo  32675  cvmlift3  32703
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