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Theorem restrcl 21765
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 21512 . . 3 ((𝐽t 𝐴) ∈ Top → ∅ ∈ (𝐽t 𝐴))
2 n0i 4282 . . 3 (∅ ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
31, 2syl 17 . 2 ((𝐽t 𝐴) ∈ Top → ¬ (𝐽t 𝐴) = ∅)
4 restfn 16698 . . . 4 t Fn (V × V)
54fndmi 6444 . . 3 dom ↾t = (V × V)
65ndmov 7326 . 2 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
73, 6nsyl2 143 1 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  c0 4276   × cxp 5540  (class class class)co 7149  t crest 16694  Topctop 21501
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-rest 16696  df-top 21502
This theorem is referenced by:  cnrest2r  21895  imacmp  22005  fiuncmp  22012  conncompss  22041  kgeni  22145  kgencmp  22153  kgencmp2  22154
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