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Theorem cvmtop2 32405
Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
Assertion
Ref Expression
cvmtop2 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)

Proof of Theorem cvmtop2
StepHypRef 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))
42, 3ax-mp 5 . . . 4 dom CovMap = (Top × Top)
54ndmov 7321 . . 3 (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
61, 5nsyl2 143 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
76simprd 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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