Theorem cvmtop2 35255

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 4306	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2		fncvm 35251	. . . . 5 ⊢ CovMap Fn (Top × Top)
3	2	fndmi 6625	. . . 4 ⊢ dom CovMap = (Top × Top)
4	3	ndmov 7576	. . 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 1540   ∈ wcel 2109  ∅c0 4299   × cxp 5639  (class class class)co 7390  Topctop 22787   CovMap ccvm 35249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-cvm 35250

This theorem is referenced by:  cvmsf1o  35266  cvmsss2  35268  cvmcov2  35269  cvmopnlem  35272  cvmliftlem8  35286  cvmlift3lem9  35321