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Theorem restin 16897
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 4517 . . . . 5  |-  ( J  e.  V  ->  U. J  e.  _V )
31, 2syl5eqel 2367 . . . 4  |-  ( J  e.  V  ->  X  e.  _V )
43adantr 451 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  X  e.  _V )
5 restco 16895 . . . 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 13338 . . . 4  |-  ( J  e.  V  ->  ( Jt  X )  =  J )
98adantr 451 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  X )  =  J )
109oveq1d 5873 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  A ) )
11 incom 3361 . . . 4  |-  ( X  i^i  A )  =  ( A  i^i  X
)
1211oveq2i 5869 . . 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 2323 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   U.cuni 3827  (class class class)co 5858   ↾t crest 13325
This theorem is referenced by:  restuni2  16898  cnrest2r  17015  cnrmi  17088  restcnrm  17090  resthauslem  17091  imacmp  17124  fiuncmp  17131  kgeni  17232  ressxms  18071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-rest 13327
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