Theorem cvmtop2 35259

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 4349	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2		fncvm 35255	. . . . 5 ⊢ CovMap Fn (Top × Top)
3	2	fndmi 6680	. . . 4 ⊢ dom CovMap = (Top × Top)
4	3	ndmov 7624	. . 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 1539   ∈ wcel 2108  ∅c0 4342   × cxp 5691  (class class class)co 7438  Topctop 22924   CovMap ccvm 35253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-1st 8022  df-2nd 8023  df-cvm 35254

This theorem is referenced by:  cvmsf1o  35270  cvmsss2  35272  cvmcov2  35273  cvmopnlem  35276  cvmliftlem8  35290  cvmlift3lem9  35325