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Theorem cvmtop1 32410
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 4303 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2 fncvm 32407 . . . . 5 CovMap Fn (Top × Top)
3 fndm 6454 . . . . 5 ( CovMap Fn (Top × Top) → dom CovMap = (Top × Top))
42, 3ax-mp 5 . . . 4 dom CovMap = (Top × Top)
54ndmov 7326 . . 3 (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
61, 5nsyl2 143 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
76simpld 495 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  c0 4295   × cxp 5552  dom cdm 5554   Fn wfn 6349  (class class class)co 7150  Topctop 21436   CovMap ccvm 32405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-cvm 32406
This theorem is referenced by:  cvmsf1o  32422  cvmscld  32423  cvmsss2  32424  cvmopnlem  32428  cvmliftmolem1  32431  cvmliftlem8  32442  cvmlift2lem9a  32453  cvmlift2lem9  32461  cvmlift2lem11  32463  cvmlift2lem12  32464  cvmliftphtlem  32467  cvmlift3lem6  32474  cvmlift3lem8  32476  cvmlift3lem9  32477
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