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Theorem cvmtop1 31227
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
StepHypRef Expression
1 n0i 3918 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → ¬ (𝐶 CovMap 𝐽) = ∅)
2 fncvm 31224 . . . . 5 CovMap Fn (Top × Top)
3 fndm 5988 . . . . 5 ( CovMap Fn (Top × Top) → dom CovMap = (Top × Top))
42, 3ax-mp 5 . . . 4 dom CovMap = (Top × Top)
54ndmov 6815 . . 3 (¬ (𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = ∅)
61, 5nsyl2 142 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
76simpld 475 1 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  c0 3913   × cxp 5110  dom cdm 5112   Fn wfn 5881  (class class class)co 6647  Topctop 20692   CovMap ccvm 31222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-cvm 31223
This theorem is referenced by:  cvmsf1o  31239  cvmscld  31240  cvmsss2  31241  cvmopnlem  31245  cvmliftmolem1  31248  cvmliftlem8  31259  cvmlift2lem9a  31270  cvmlift2lem9  31278  cvmlift2lem11  31280  cvmlift2lem12  31281  cvmliftphtlem  31284  cvmlift3lem6  31291  cvmlift3lem8  31293  cvmlift3lem9  31294
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