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Theorem subspopn 26356
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 13619 . . . 4  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  i^i  A
)  e.  ( Jt  A ) )
2 df-ss 3302 . . . . 5  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
3 eleq1 2472 . . . . 5  |-  ( ( B  i^i  A )  =  B  ->  (
( B  i^i  A
)  e.  ( Jt  A )  <->  B  e.  ( Jt  A ) ) )
42, 3sylbi 188 . . . 4  |-  ( B 
C_  A  ->  (
( B  i^i  A
)  e.  ( Jt  A )  <->  B  e.  ( Jt  A ) ) )
51, 4syl5ibcom 212 . . 3  |-  ( ( J  e.  Top  /\  A  e.  V  /\  B  e.  J )  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
653expa 1153 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  B  e.  J
)  ->  ( B  C_  A  ->  B  e.  ( Jt  A ) ) )
76impr 603 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288  (class class class)co 6048   ↾t crest 13611   Topctop 16921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-rest 13613
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