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Theorem restopnb 19521
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 1004 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  B )
2 simpr2 1003 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  B  C_  A )
31, 2sstrd 3519 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  A )
4 df-ss 3495 . . . . . 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 2475 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  =  ( C  i^i  A ) )
7 ineq1 3698 . . . . . . 7  |-  ( v  =  C  ->  (
v  i^i  A )  =  ( C  i^i  A ) )
87eqeq2d 2481 . . . . . 6  |-  ( v  =  C  ->  ( C  =  ( v  i^i  A )  <->  C  =  ( C  i^i  A ) ) )
98rspcev 3219 . . . . 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 3713 . . . . . 6  |-  ( ( v  i^i  A )  i^i  B )  =  ( v  i^i  ( A  i^i  B ) )
13 simprr 756 . . . . . . . 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 3704 . . . . . . 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 1040 . . . . . . . . 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 3495 . . . . . . . . 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 716 . . . . . . 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 2510 . . . . . 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 1039 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  B  C_  A )
21 dfss1 3708 . . . . . . . . 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 3705 . . . . . . 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 716 . . . . . 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 2532 . . . . 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 757 . . . . . 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 755 . . . . . 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 1038 . . . . . 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 19254 . . . . . 6  |-  ( ( J  e.  Top  /\  v  e.  J  /\  B  e.  J )  ->  ( v  i^i  B
)  e.  J )
3026, 27, 28, 29syl3anc 1228 . . . . 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 2555 . . . 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 2960 . . 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 14695 . . 3  |-  ( ( J  e.  Top  /\  A  e.  V )  ->  ( C  e.  ( Jt  A )  <->  E. v  e.  J  C  =  ( v  i^i  A
) ) )
3534adantr 465 . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2818    i^i cin 3480    C_ wss 3481  (class class class)co 6294   ↾t crest 14688   Topctop 19240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-rest 14690  df-top 19245
This theorem is referenced by:  restopn2  19523  cxpcn3  22965  pnfneige0  27726  fourierdlem62  31760  fouriersw  31823
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