Theorem cvmtop1 35994

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 4286	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2		fncvm 35991	. . . . 5 ⊢ CovMap Fn (Top × Top)
3	2	fndmi 6635	. . . 4 ⊢ dom CovMap = (Top × Top)
4	3	ndmov 7597	. . 3 ⊢ (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
5	1, 4	nsyl2 142	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6	5	simpld 500	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∅c0 4279   × cxp 5649  (class class class)co 7412  Topctop 23191   CovMap ccvm 35989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-cvm 35990

This theorem is used by:  cvmsf1o  36006  cvmscld  36007  cvmsss2  36008  cvmopnlem  36012  cvmliftmolem1  36015  cvmliftlem8  36026  cvmlift2lem9a  36037  cvmlift2lem9  36045  cvmlift2lem11  36047  cvmlift2lem12  36048  cvmliftphtlem  36051  cvmlift3lem6  36058  cvmlift3lem8  36060  cvmlift3lem9  36061