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Theorem cvmsi 35271
Description: One direction of cvmsval 35272. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
cvmcov.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmsi (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑇,𝑠,𝑢,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmsi
StepHypRef Expression
1 cvmcov.1 . . 3 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
21cvmsrcl 35270 . 2 (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)
3 imaeq2 6073 . . . . . . . . . . 11 (𝑘 = 𝑈 → (𝐹𝑘) = (𝐹𝑈))
43eqeq2d 2747 . . . . . . . . . 10 (𝑘 = 𝑈 → ( 𝑠 = (𝐹𝑘) ↔ 𝑠 = (𝐹𝑈)))
5 oveq2 7440 . . . . . . . . . . . . . 14 (𝑘 = 𝑈 → (𝐽t 𝑘) = (𝐽t 𝑈))
65oveq2d 7448 . . . . . . . . . . . . 13 (𝑘 = 𝑈 → ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))
76eleq2d 2826 . . . . . . . . . . . 12 (𝑘 = 𝑈 → ((𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
87anbi2d 630 . . . . . . . . . . 11 (𝑘 = 𝑈 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
98ralbidv 3177 . . . . . . . . . 10 (𝑘 = 𝑈 → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
104, 9anbi12d 632 . . . . . . . . 9 (𝑘 = 𝑈 → (( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))) ↔ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
1110rabbidv 3443 . . . . . . . 8 (𝑘 = 𝑈 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))})
1211, 1fvmptss2 7041 . . . . . . 7 (𝑆𝑈) ⊆ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))}
1312sseli 3978 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))})
14 unieq 4917 . . . . . . . . 9 (𝑠 = 𝑇 𝑠 = 𝑇)
1514eqeq1d 2738 . . . . . . . 8 (𝑠 = 𝑇 → ( 𝑠 = (𝐹𝑈) ↔ 𝑇 = (𝐹𝑈)))
16 difeq1 4118 . . . . . . . . . . 11 (𝑠 = 𝑇 → (𝑠 ∖ {𝑢}) = (𝑇 ∖ {𝑢}))
1716raleqdv 3325 . . . . . . . . . 10 (𝑠 = 𝑇 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅))
1817anbi1d 631 . . . . . . . . 9 (𝑠 = 𝑇 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) ↔ (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
1918raleqbi1dv 3337 . . . . . . . 8 (𝑠 = 𝑇 → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) ↔ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
2015, 19anbi12d 632 . . . . . . 7 (𝑠 = 𝑇 → (( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))) ↔ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
2120elrab 3691 . . . . . 6 (𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))} ↔ (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
2213, 21sylib 218 . . . . 5 (𝑇 ∈ (𝑆𝑈) → (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
2322simpld 494 . . . 4 (𝑇 ∈ (𝑆𝑈) → 𝑇 ∈ (𝒫 𝐶 ∖ {∅}))
24 eldifsn 4785 . . . 4 (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅))
2523, 24sylib 218 . . 3 (𝑇 ∈ (𝑆𝑈) → (𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅))
26 elpwi 4606 . . . 4 (𝑇 ∈ 𝒫 𝐶𝑇𝐶)
2726anim1i 615 . . 3 ((𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅) → (𝑇𝐶𝑇 ≠ ∅))
2825, 27syl 17 . 2 (𝑇 ∈ (𝑆𝑈) → (𝑇𝐶𝑇 ≠ ∅))
2922simprd 495 . 2 (𝑇 ∈ (𝑆𝑈) → ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
302, 28, 293jca 1128 1 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  {crab 3435  cdif 3947  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   cuni 4906  cmpt 5224  ccnv 5683  cres 5686  cima 5687  cfv 6560  (class class class)co 7432  t crest 17466  Homeochmeo 23762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435
This theorem is referenced by:  cvmsval  35272  cvmsss  35273  cvmsn0  35274  cvmsuni  35275  cvmsdisj  35276  cvmshmeo  35277
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