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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmsrcl | Structured version Visualization version GIF version |
Description: Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.) |
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))))}) |
Ref | Expression |
---|---|
cvmsrcl | ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) |
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) |
Colors of variables: wff setvar class |
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 |
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