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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmtop1 | Structured version Visualization version GIF version |
Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
cvmtop1 | ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4299 | . . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅) | |
2 | fncvm 32504 | . . . . 5 ⊢ CovMap Fn (Top × Top) | |
3 | fndm 6455 | . . . . 5 ⊢ ( CovMap Fn (Top × Top) → dom CovMap = (Top × Top)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom CovMap = (Top × Top) |
5 | 4 | ndmov 7332 | . . 3 ⊢ (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅) |
6 | 1, 5 | nsyl2 143 | . 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top)) |
7 | 6 | simpld 497 | 1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4291 × cxp 5553 dom cdm 5555 Fn wfn 6350 (class class class)co 7156 Topctop 21501 CovMap ccvm 32502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-cvm 32503 |
This theorem is referenced by: cvmsf1o 32519 cvmscld 32520 cvmsss2 32521 cvmopnlem 32525 cvmliftmolem1 32528 cvmliftlem8 32539 cvmlift2lem9a 32550 cvmlift2lem9 32558 cvmlift2lem11 32560 cvmlift2lem12 32561 cvmliftphtlem 32564 cvmlift3lem6 32571 cvmlift3lem8 32573 cvmlift3lem9 32574 |
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