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Theorem restrcl 14064
Description: Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
Assertion
Ref Expression
restrcl  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)

Proof of Theorem restrcl
Dummy variables  x  j  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0opn 13903 . 2  |-  ( ( Jt  A )  e.  Top  -> 
(/)  e.  ( Jt  A
) )
2 df-rest 12712 . . 3  |-t  =  ( j  e.  _V ,  x  e. 
_V  |->  ran  ( y  e.  j  |->  ( y  i^i  x ) ) )
32elmpocl 6086 . 2  |-  ( (/)  e.  ( Jt  A )  ->  ( J  e.  _V  /\  A  e.  _V ) )
41, 3syl 14 1  |-  ( ( Jt  A )  e.  Top  ->  ( J  e.  _V  /\  A  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   _Vcvv 2752    i^i cin 3143   (/)c0 3437    |-> cmpt 4079   ran crn 4642  (class class class)co 5891   ↾t crest 12710   Topctop 13894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-rest 12712  df-top 13895
This theorem is referenced by:  cnrest2r  14134
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