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Theorem ssrest 13575
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  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 110 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Jt  A
) )
2 ssrexv 3220 . . . . . 6  |-  ( J 
C_  K  ->  ( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
32ad2antlr 489 . . . . 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 df-rest 12680 . . . . . . . 8  |-t  =  ( j  e.  _V ,  x  e. 
_V  |->  ran  ( y  e.  j  |->  ( y  i^i  x ) ) )
54elmpocl 6068 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
65adantl 277 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( J  e.  _V  /\  A  e.  _V )
)
7 elrest 12685 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
86, 7syl 14 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
9 simpll 527 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  K  e.  V )
106simprd 114 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  A  e.  _V )
11 elrest 12685 . . . . . 6  |-  ( ( K  e.  V  /\  A  e.  _V )  ->  ( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
129, 10, 11syl2anc 411 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
133, 8, 123imtr4d 203 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
141, 13mpd 13 . . 3  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Kt  A
) )
1514ex 115 . 2  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
1615ssrdv 3161 1  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   _Vcvv 2737    i^i cin 3128    C_ wss 3129    |-> cmpt 4064   ran crn 4627  (class class class)co 5874   ↾t crest 12678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-rest 12680
This theorem is referenced by: (None)
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