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Theorem subspopn 31507
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 15041 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
2 df-ss 3427 . . . . 5  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
3 eleq1 2474 . . . . 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 1197 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  B  e.  J
)  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
76impr 617 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413  (class class class)co 6277   ↾t crest 15033   Topctop 19684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-rest 15035
This theorem is referenced by: (None)
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