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Theorem subspopn 26471
Description: An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
subspopn  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A ) )  ->  B  e.  ( Jt  A
) )

Proof of Theorem subspopn
StepHypRef Expression
1 elrestr 13661 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
2 df-ss 3336 . . . . 5  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
3 eleq1 2498 . . . . 5  |-  ( ( B  i^i  A )  =  B  ->  (
( B  i^i  A
)  e.  ( Jt  A )  <->  B  e.  ( Jt  A ) ) )
42, 3sylbi 189 . . . 4  |-  ( B 
C_  A  ->  (
( B  i^i  A
)  e.  ( Jt  A )  <->  B  e.  ( Jt  A ) ) )
51, 4syl5ibcom 213 . . 3  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
653expa 1154 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  B  e.  J
)  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
76impr 604 1  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A ) )  ->  B  e.  ( Jt  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322  (class class class)co 6084   ↾t crest 13653   Topctop 16963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-rest 13655
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