MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restopnb Structured version   Unicode version

Theorem restopnb 18779
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 996 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  B )
2 simpr2 995 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  B  C_  A )
31, 2sstrd 3366 . . . . . 6  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  C_  A )
4 df-ss 3342 . . . . . 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 2448 . . . 4  |-  ( ( ( J  e.  Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B
) )  ->  C  =  ( C  i^i  A ) )
7 ineq1 3545 . . . . . . 7  |-  ( v  =  C  ->  (
v  i^i  A )  =  ( C  i^i  A ) )
87eqeq2d 2454 . . . . . 6  |-  ( v  =  C  ->  ( C  =  ( v  i^i  A )  <->  C  =  ( C  i^i  A ) ) )
98rspcev 3073 . . . . 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 3560 . . . . . 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 3551 . . . . . . 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 1032 . . . . . . . . 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 3342 . . . . . . . . 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 2477 . . . . . 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 1031 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  A  e.  V )  /\  ( B  e.  J  /\  B  C_  A  /\  C  C_  B ) )  /\  v  e.  J )  ->  B  C_  A )
21 dfss1 3555 . . . . . . . . 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 3552 . . . . . . 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 2499 . . . . 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 1030 . . . . . 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 18512 . . . . . 6  |-  ( ( J  e.  Top  /\  v  e.  J  /\  B  e.  J )  ->  ( v  i^i  B
)  e.  J )
3026, 27, 28, 29syl3anc 1218 . . . . 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 2517 . . . 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 2842 . . 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 14366 . . 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 965    = wceq 1369    e. wcel 1756   E.wrex 2716    i^i cin 3327    C_ wss 3328  (class class class)co 6091   ↾t crest 14359   Topctop 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-rest 14361  df-top 18503
This theorem is referenced by:  restopn2  18781  cxpcn3  22186  pnfneige0  26381
  Copyright terms: Public domain W3C validator