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Theorem restrcl 22306
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 22051 . . 3 ((𝐽t 𝐴) ∈ Top → ∅ ∈ (𝐽t 𝐴))
2 n0i 4273 . . 3 (∅ ∈ (𝐽t 𝐴) → ¬ (𝐽t 𝐴) = ∅)
31, 2syl 17 . 2 ((𝐽t 𝐴) ∈ Top → ¬ (𝐽t 𝐴) = ∅)
4 restfn 17133 . . . 4 t Fn (V × V)
54fndmi 6535 . . 3 dom ↾t = (V × V)
65ndmov 7450 . 2 (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ∅)
73, 6nsyl2 141 1 ((𝐽t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1542  wcel 2110  Vcvv 3431  c0 4262   × cxp 5588  (class class class)co 7271  t crest 17129  Topctop 22040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-rest 17131  df-top 22041
This theorem is referenced by:  cnrest2r  22436  imacmp  22546  fiuncmp  22553  conncompss  22582  kgeni  22686  kgencmp  22694  kgencmp2  22695
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