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Theorem restopnb 18679
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 991 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  B )
2 simpr2 990 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  B  C_  A )
31, 2sstrd 3363 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  A )
4 df-ss 3339 . . . . . 6  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
53, 4sylib 196 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  ( C  i^i  A )  =  C )
65eqcomd 2446 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  =  ( C  i^i  A ) )
7 ineq1 3542 . . . . . . 7  |-  ( v  =  C  ->  (
v  i^i  A )  =  ( C  i^i  A ) )
87eqeq2d 2452 . . . . . 6  |-  ( v  =  C  ->  ( C  =  ( v  i^i  A )  <->  C  =  ( C  i^i  A ) ) )
98rspcev 3070 . . . . 5  |-  ( ( C  e.  J  /\  C  =  ( C  i^i  A ) )  ->  E. v  e.  J  C  =  ( v  i^i  A ) )
109expcom 435 . . . 4  |-  ( C  =  ( C  i^i  A )  ->  ( C  e.  J  ->  E. v  e.  J  C  =  ( v  i^i  A
) ) )
116, 10syl 16 . . 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 3557 . . . . . 6  |-  ( ( v  i^i  A )  i^i  B )  =  ( v  i^i  ( A  i^i  B ) )
13 simprr 751 . . . . . . . 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  =  ( v  i^i 
A ) )
1413ineq1d 3548 . . . . . . 7  |-  ( ( ( ( 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 1027 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  C  C_  B )
16 df-ss 3339 . . . . . . . . 9  |-  ( C 
C_  B  <->  ( C  i^i  B )  =  C )
1715, 16sylib 196 . . . . . . . 8  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  ( C  i^i  B
)  =  C )
1817adantrr 711 . . . . . . 7  |-  ( ( ( ( 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 2475 . . . . . 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  A
)  i^i  B )  =  C )
20 simplr2 1026 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  B  C_  A )
21 dfss1 3552 . . . . . . . . 9  |-  ( B 
C_  A  <->  ( A  i^i  B )  =  B )
2220, 21sylib 196 . . . . . . . 8  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  ( A  i^i  B
)  =  B )
2322ineq2d 3549 . . . . . . 7  |-  ( ( ( ( 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 711 . . . . . 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  ( A  i^i  B ) )  =  ( v  i^i  B
) )
2512, 19, 243eqtr3a 2497 . . . . 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  =  ( v  i^i 
B ) )
26 simplll 752 . . . . . 6  |-  ( ( ( ( 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 750 . . . . . 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  e.  J )
28 simplr1 1025 . . . . . 6  |-  ( ( ( ( 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 18412 . . . . . 6  |-  ( ( J  e.  Top  /\  v  e.  J  /\  B  e.  J )  ->  ( v  i^i  B
)  e.  J )
3026, 27, 28, 29syl3anc 1213 . . . . 5  |-  ( ( ( ( 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 2515 . . . 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 )
3231rexlimdvaa 2840 . . 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 ) )
3311, 32impbid 191 . 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
) ) )
34 elrest 14362 . . 3  |-  ( ( J  e.  Top  /\  A  e.  V )  ->  ( C  e.  ( Jt  A )  <->  E. v  e.  J  C  =  ( v  i^i  A
) ) )
3534adantr 462 . 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 ) ) )
3633, 35bitr4d 256 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714    i^i cin 3324    C_ wss 3325  (class class class)co 6090   ↾t crest 14355   Topctop 18398
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-rest 14357  df-top 18403
This theorem is referenced by:  restopn2  18681  cxpcn3  22129  pnfneige0  26301
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