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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmtop2 | Structured version Visualization version GIF version |
Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.) |
Ref | Expression |
---|---|
cvmtop2 | ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4296 | . . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅) | |
2 | fncvm 32401 | . . . . 5 ⊢ CovMap Fn (Top × Top) | |
3 | fndm 6448 | . . . . 5 ⊢ ( CovMap Fn (Top × Top) → dom CovMap = (Top × Top)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom CovMap = (Top × Top) |
5 | 4 | ndmov 7321 | . . 3 ⊢ (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅) |
6 | 1, 5 | nsyl2 143 | . 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top)) |
7 | 6 | simprd 496 | 1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 × cxp 5546 dom cdm 5548 Fn wfn 6343 (class class class)co 7145 Topctop 21429 CovMap ccvm 32399 |
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 ax-un 7450 |
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-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 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-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-cvm 32400 |
This theorem is referenced by: cvmsf1o 32416 cvmsss2 32418 cvmcov2 32419 cvmopnlem 32422 cvmliftlem8 32436 cvmlift3lem9 32471 |
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