Theorem cvmtop1 32620

Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
cvmtop1	⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)

Proof of Theorem cvmtop1

Step	Hyp	Ref	Expression
1		n0i 4249	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2		fncvm 32617	. . . . 5 ⊢ CovMap Fn (Top × Top)
3	2	fndmi 6426	. . . 4 ⊢ dom CovMap = (Top × Top)
4	3	ndmov 7312	. . 3 ⊢ (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
5	1, 4	nsyl2 143	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6	5	simpld 498	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4243   × cxp 5517  (class class class)co 7135  Topctop 21498   CovMap ccvm 32615

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-cvm 32616

This theorem is referenced by:  cvmsf1o  32632  cvmscld  32633  cvmsss2  32634  cvmopnlem  32638  cvmliftmolem1  32641  cvmliftlem8  32652  cvmlift2lem9a  32663  cvmlift2lem9  32671  cvmlift2lem11  32673  cvmlift2lem12  32674  cvmliftphtlem  32677  cvmlift3lem6  32684  cvmlift3lem8  32686  cvmlift3lem9  32687