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Theorem restrcl 20942
Description: Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restrcl ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))

Proof of Theorem restrcl
StepHypRef Expression
1 0opn 20690 . . 3 ((𝐽t 𝐴) ∈ Top → ∅ ∈ (𝐽t 𝐴))
2 n0i 3912 . . 3 (∅ ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
31, 2syl 17 . 2 ((𝐽t 𝐴) ∈ Top → ¬ (𝐽t 𝐴) = ∅)
4 restfn 16066 . . . 4 t Fn (V × V)
5 fndm 5978 . . . 4 ( ↾t Fn (V × V) → dom ↾t = (V × V))
64, 5ax-mp 5 . . 3 dom ↾t = (V × V)
76ndmov 6803 . 2 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
83, 7nsyl2 142 1 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907   × cxp 5102  dom cdm 5104   Fn wfn 5871  (class class class)co 6635  t crest 16062  Topctop 20679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-rest 16064  df-top 20680
This theorem is referenced by:  cnrest2r  21072  imacmp  21181  fiuncmp  21188  conncompss  21217  kgeni  21321  kgencmp  21329  kgencmp2  21330
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