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Theorem restrcl 13894
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 13733 . 2  |-  ( ( Jt  A )  e.  Top  -> 
(/)  e.  ( Jt  A
) )
2 df-rest 12707 . . 3  |-t  =  ( j  e.  _V ,  x  e. 
_V  |->  ran  ( y  e.  j  |->  ( y  i^i  x ) ) )
32elmpocl 6082 . 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 2158   _Vcvv 2749    i^i cin 3140   (/)c0 3434    |-> cmpt 4076   ran crn 4639  (class class class)co 5888   ↾t crest 12705   Topctop 13724
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-rest 12707  df-top 13725
This theorem is referenced by:  cnrest2r  13964
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