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

Theorem ssrest 19844
Description: If  K is a finer topology than  J, then the subspace topologies induced by  A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
ssrest  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )

Proof of Theorem ssrest
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 459 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Jt  A
) )
2 ssrexv 3551 . . . . . 6  |-  ( J 
C_  K  ->  ( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
32ad2antlr 724 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A ) ) )
4 n0i 3788 . . . . . . . 8  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
5 restfn 14914 . . . . . . . . . 10  |-t  Fn  ( _V  X.  _V )
6 fndm 5662 . . . . . . . . . 10  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
75, 6ax-mp 5 . . . . . . . . 9  |-  domt  =  ( _V  X.  _V )
87ndmov 6432 . . . . . . . 8  |-  ( -.  ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  (/) )
94, 8nsyl2 127 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
109adantl 464 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( J  e.  _V  /\  A  e.  _V )
)
11 elrest 14917 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
1210, 11syl 16 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
13 simpll 751 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  K  e.  V )
1410simprd 461 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  A  e.  _V )
15 elrest 14917 . . . . . 6  |-  ( ( K  e.  V  /\  A  e.  _V )  ->  ( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
1613, 14, 15syl2anc 659 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
173, 12, 163imtr4d 268 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
181, 17mpd 15 . . 3  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Kt  A
) )
1918ex 432 . 2  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
2019ssrdv 3495 1  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   (/)c0 3783    X. cxp 4986   dom cdm 4988    Fn wfn 5565  (class class class)co 6270   ↾t crest 14910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-rest 14912
This theorem is referenced by:  1stcrest  20120  kgencmp  20212  kgencmp2  20213  kgen2ss  20222  ssufl  20585
  Copyright terms: Public domain W3C validator