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Theorem cvmtop2 33855
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 4293 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2 fncvm 33851 . . . . 5 CovMap Fn (Top × Top)
32fndmi 6606 . . . 4 dom CovMap = (Top × Top)
43ndmov 7538 . . 3 (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
51, 4nsyl2 141 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
65simprd 496 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  c0 4282   × cxp 5631  (class class class)co 7357  Topctop 22242   CovMap ccvm 33849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-cvm 33850
This theorem is referenced by:  cvmsf1o  33866  cvmsss2  33868  cvmcov2  33869  cvmopnlem  33872  cvmliftlem8  33886  cvmlift3lem9  33921
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