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Theorem restin 16913
Description: When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)
Hypothesis
Ref Expression
restin.1  |-  X  = 
U. J
Assertion
Ref Expression
restin  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  X
) ) )

Proof of Theorem restin
StepHypRef Expression
1 restin.1 . . . . 5  |-  X  = 
U. J
2 uniexg 4533 . . . . 5  |-  ( J  e.  V  ->  U. J  e.  _V )
31, 2syl5eqel 2380 . . . 4  |-  ( J  e.  V  ->  X  e.  _V )
43adantr 451 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  X  e.  _V )
5 restco 16911 . . . 4  |-  ( ( J  e.  V  /\  X  e.  _V  /\  A  e.  W )  ->  (
( Jt  X )t  A )  =  ( Jt  ( X  i^i  A
) ) )
653com23 1157 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  X  e.  _V )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
74, 6mpd3an3 1278 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
81restid 13354 . . . 4  |-  ( J  e.  V  ->  ( Jt  X )  =  J )
98adantr 451 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  X )  =  J )
109oveq1d 5889 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  A ) )
11 incom 3374 . . . 4  |-  ( X  i^i  A )  =  ( A  i^i  X
)
1211oveq2i 5885 . . 3  |-  ( Jt  ( X  i^i  A ) )  =  ( Jt  ( A  i^i  X ) )
1312a1i 10 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  ( X  i^i  A ) )  =  ( Jt  ( A  i^i  X
) ) )
147, 10, 133eqtr3d 2336 1  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   U.cuni 3843  (class class class)co 5874   ↾t crest 13341
This theorem is referenced by:  restuni2  16914  cnrest2r  17031  cnrmi  17104  restcnrm  17106  resthauslem  17107  imacmp  17140  fiuncmp  17147  kgeni  17248  ressxms  18087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-rest 13343
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