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Theorem restopnb 16906
Description: If  B is an open subset of the subspace base set  A, then any subset of  B is open iff it is open in  A. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
restopnb  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  e.  J  <->  C  e.  ( Jt  A ) ) )

Proof of Theorem restopnb
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simpr3 963 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  B )
2 simpr2 962 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  B  C_  A )
31, 2sstrd 3189 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  A )
4 df-ss 3166 . . . . . 6  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
53, 4sylib 188 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  i^i  A )  =  C )
65eqcomd 2288 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  =  ( C  i^i  A ) )
7 ineq1 3363 . . . . . . 7  |-  ( v  =  C  ->  (
v  i^i  A )  =  ( C  i^i  A ) )
87eqeq2d 2294 . . . . . 6  |-  ( v  =  C  ->  ( C  =  ( v  i^i  A )  <->  C  =  ( C  i^i  A ) ) )
98rspcev 2884 . . . . 5  |-  ( ( C  e.  J  /\  C  =  ( C  i^i  A ) )  ->  E. v  e.  J  C  =  ( v  i^i  A ) )
109expcom 424 . . . 4  |-  ( C  =  ( C  i^i  A )  ->  ( C  e.  J  ->  E. v  e.  J  C  =  ( v  i^i  A
) ) )
116, 10syl 15 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  e.  J  ->  E. v  e.  J  C  =  ( v  i^i 
A ) ) )
12 inass 3379 . . . . . . 7  |-  ( ( v  i^i  A )  i^i  B )  =  ( v  i^i  ( A  i^i  B ) )
13 simprr 733 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  C  =  ( v  i^i 
A ) )
1413ineq1d 3369 . . . . . . . 8  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  ( C  i^i  B )  =  ( ( v  i^i 
A )  i^i  B
) )
15 simplr3 999 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  C  C_  B )
16 df-ss 3166 . . . . . . . . . 10  |-  ( C 
C_  B  <->  ( C  i^i  B )  =  C )
1715, 16sylib 188 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  ( C  i^i  B
)  =  C )
1817adantrr 697 . . . . . . . 8  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  ( C  i^i  B )  =  C )
1914, 18eqtr3d 2317 . . . . . . 7  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  (
( v  i^i  A
)  i^i  B )  =  C )
20 simplr2 998 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  B  C_  A )
21 dfss1 3373 . . . . . . . . . 10  |-  ( B 
C_  A  <->  ( A  i^i  B )  =  B )
2220, 21sylib 188 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  ( A  i^i  B
)  =  B )
2322ineq2d 3370 . . . . . . . 8  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  ( v  i^i  ( A  i^i  B ) )  =  ( v  i^i 
B ) )
2423adantrr 697 . . . . . . 7  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  (
v  i^i  ( A  i^i  B ) )  =  ( v  i^i  B
) )
2512, 19, 243eqtr3a 2339 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  C  =  ( v  i^i 
B ) )
26 simplll 734 . . . . . . 7  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  J  e.  Top )
27 simprl 732 . . . . . . 7  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  v  e.  J )
28 simplr1 997 . . . . . . 7  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  B  e.  J )
29 inopn 16645 . . . . . . 7  |-  ( ( J  e.  Top  /\  v  e.  J  /\  B  e.  J )  ->  ( v  i^i  B
)  e.  J )
3026, 27, 28, 29syl3anc 1182 . . . . . 6  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  (
v  i^i  B )  e.  J )
3125, 30eqeltrd 2357 . . . . 5  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  ( v  e.  J  /\  C  =  (
v  i^i  A )
) )  ->  C  e.  J )
3231expr 598 . . . 4  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  ( C  =  ( v  i^i  A )  ->  C  e.  J
) )
3332rexlimdva 2667 . . 3  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( E. v  e.  J  C  =  ( v  i^i  A )  ->  C  e.  J ) )
3411, 33impbid 183 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  e.  J  <->  E. v  e.  J  C  =  ( v  i^i  A
) ) )
35 elrest 13332 . . 3  |-  ( ( J  e.  Top  /\  A  e.  V )  ->  ( C  e.  ( Jt  A )  <->  E. v  e.  J  C  =  ( v  i^i  A
) ) )
3635adantr 451 . 2  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  e.  ( Jt  A
)  <->  E. v  e.  J  C  =  ( v  i^i  A ) ) )
3734, 36bitr4d 247 1  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  e.  J  <->  C  e.  ( Jt  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152  (class class class)co 5858   ↾t crest 13325   Topctop 16631
This theorem is referenced by:  restopn2  16908  cxpcn3  20088  pnfneige0  23374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-rest 13327  df-top 16636
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