Theorem restrcl 15191

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	|- ( ( J ↾t 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.

Step	Hyp	Ref	Expression
1		0opn 15030	. 2 |- ( ( J ↾t A ) e. Top -> (/) e. ( J ↾t A ) )
2		df-rest 13572	. . 3 |- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
3	2	elmpocl 6274	. 2 |- ( (/) e. ( J ↾t A ) -> ( J e. _V /\ A e. _V ) )
4	1, 3	syl 14	1 |- ( ( J ↾t A ) e. Top -> ( J e. _V /\ A e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   i^i cin 3219   (/)c0 3520   |-> cmpt 4187   ran crn 4770  (class class class)co 6075   ↾_t crest 13570   Topctop 15021

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-rest 13572  df-top 15022

This theorem is referenced by:  cnrest2r  15261