Theorem cvmsrcl 35270

Description: Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypothesis

Ref	Expression
cvmcov.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
cvmsrcl	⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)

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 cvmsrcl

Step	Hyp	Ref	Expression
1		cvmcov.1	. . 3 ⊢ 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	1	dmmptss 6260	. 2 ⊢ dom 𝑆 ⊆ 𝐽
3		elfvdm 6942	. 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ dom 𝑆)
4	2, 3	sselid 3980	1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  {crab 3435   ∖ cdif 3947   ∩ cin 3949  ∅c0 4332  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906   ↦ cmpt 5224  ^◡ccnv 5683  dom cdm 5684   ↾ 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-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-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fv 6568

This theorem is referenced by:  cvmsi  35271  cvmsf1o  35278  cvmsss2  35280  cvmopnlem  35284  cvmliftlem8  35298  cvmlift2lem9  35317  cvmlift2lem10  35318  cvmlift3lem6  35330  cvmlift3lem8  35332