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Theorem cvmtop1 31580
Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
cvmtop1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)

Proof of Theorem cvmtop1
StepHypRef Expression
1 n0i 4068 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2 fncvm 31577 . . . . 5 CovMap Fn (Top × Top)
3 fndm 6130 . . . . 5 ( CovMap Fn (Top × Top) → dom CovMap = (Top × Top))
42, 3ax-mp 5 . . . 4 dom CovMap = (Top × Top)
54ndmov 6965 . . 3 (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
61, 5nsyl2 144 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
76simpld 482 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  c0 4063   × cxp 5247  dom cdm 5249   Fn wfn 6026  (class class class)co 6793  Topctop 20918   CovMap ccvm 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-cvm 31576
This theorem is referenced by:  cvmsf1o  31592  cvmscld  31593  cvmsss2  31594  cvmopnlem  31598  cvmliftmolem1  31601  cvmliftlem8  31612  cvmlift2lem9a  31623  cvmlift2lem9  31631  cvmlift2lem11  31633  cvmlift2lem12  31634  cvmliftphtlem  31637  cvmlift3lem6  31644  cvmlift3lem8  31646  cvmlift3lem9  31647
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