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Theorem subspopn 29835
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 14673 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
2 df-ss 3483 . . . . 5  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
3 eleq1 2532 . . . . 5  |-  ( ( B  i^i  A )  =  B  ->  (
( B  i^i  A
)  e.  ( Jt  A )  <->  B  e.  ( Jt  A ) ) )
42, 3sylbi 195 . . . 4  |-  ( B 
C_  A  ->  (
( B  i^i  A
)  e.  ( Jt  A )  <->  B  e.  ( Jt  A ) ) )
51, 4syl5ibcom 220 . . 3  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
653expa 1191 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  B  e.  J
)  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
76impr 619 1  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A ) )  ->  B  e.  ( Jt  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469  (class class class)co 6275   ↾t crest 14665   Topctop 19154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-rest 14667
This theorem is referenced by: (None)
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