Theorem cvmtop2 35443

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

Step	Hyp	Ref	Expression
1		n0i 4280	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2		fncvm 35439	. . . . 5 ⊢ CovMap Fn (Top × Top)
3	2	fndmi 6602	. . . 4 ⊢ dom CovMap = (Top × Top)
4	3	ndmov 7551	. . 3 ⊢ (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
5	1, 4	nsyl2 141	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6	5	simprd 495	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∅c0 4273   × cxp 5629  (class class class)co 7367  Topctop 22858   CovMap ccvm 35437

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-cvm 35438

This theorem is referenced by:  cvmsf1o  35454  cvmsss2  35456  cvmcov2  35457  cvmopnlem  35460  cvmliftlem8  35474  cvmlift3lem9  35509