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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 6088 | . 2 ⊢ dom 𝑆 ⊆ 𝐽 |
3 | elfvdm 6695 | . 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ dom 𝑆) | |
4 | 2, 3 | sseldi 3962 | 1 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∖ cdif 3930 ∩ cin 3932 ∅c0 4288 𝒫 cpw 4535 {csn 4557 ∪ cuni 4830 ↦ cmpt 5137 ◡ccnv 5547 dom cdm 5548 ↾ cres 5550 “ cima 5551 ‘cfv 6348 (class class class)co 7145 ↾t crest 16682 Homeochmeo 22289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 |
This theorem is referenced by: cvmsi 32409 cvmsf1o 32416 cvmsss2 32418 cvmopnlem 32422 cvmliftlem8 32436 cvmlift2lem9 32455 cvmlift2lem10 32456 cvmlift3lem6 32468 cvmlift3lem8 32470 |
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