Theorem cvmsrcl 34086

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 6229	. 2 ⊢ dom 𝑆 ⊆ 𝐽
3		elfvdm 6915	. 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ dom 𝑆)
4	2, 3	sselid 3976	1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3431   ∖ cdif 3941   ∩ cin 3943  ∅c0 4318  𝒫 cpw 4596  {csn 4622  ∪ cuni 4901   ↦ cmpt 5224  ^◡ccnv 5668  dom cdm 5669   ↾ cres 5671   “ cima 5672  ‘cfv 6532  (class class class)co 7393   ↾_t crest 17348  Homeochmeo 23186

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fv 6540

This theorem is referenced by:  cvmsi  34087  cvmsf1o  34094  cvmsss2  34096  cvmopnlem  34100  cvmliftlem8  34114  cvmlift2lem9  34133  cvmlift2lem10  34134  cvmlift3lem6  34146  cvmlift3lem8  34148