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Theorem cvmtop2 31004
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 3902 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2 fncvm 31000 . . . . 5 CovMap Fn (Top × Top)
3 fndm 5958 . . . . 5 ( CovMap Fn (Top × Top) → dom CovMap = (Top × Top))
42, 3ax-mp 5 . . . 4 dom CovMap = (Top × Top)
54ndmov 6783 . . 3 (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
61, 5nsyl2 142 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
76simprd 479 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  c0 3897   × cxp 5082  dom cdm 5084   Fn wfn 5852  (class class class)co 6615  Topctop 20638   CovMap ccvm 30998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-cvm 30999
This theorem is referenced by:  cvmsf1o  31015  cvmsss2  31017  cvmcov2  31018  cvmopnlem  31021  cvmliftlem8  31035  cvmlift3lem9  31070
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