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Theorem restin 18729
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 6376 . . . . 5  |-  ( J  e.  V  ->  U. J  e.  _V )
31, 2syl5eqel 2525 . . . 4  |-  ( J  e.  V  ->  X  e.  _V )
43adantr 462 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  X  e.  _V )
5 restco 18727 . . . 4  |-  ( ( J  e.  V  /\  X  e.  _V  /\  A  e.  W )  ->  (
( Jt  X )t  A )  =  ( Jt  ( X  i^i  A
) ) )
653com23 1188 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  X  e.  _V )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
74, 6mpd3an3 1310 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
81restid 14368 . . . 4  |-  ( J  e.  V  ->  ( Jt  X )  =  J )
98adantr 462 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  X )  =  J )
109oveq1d 6105 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  A ) )
11 incom 3540 . . . 4  |-  ( X  i^i  A )  =  ( A  i^i  X
)
1211oveq2i 6101 . . 3  |-  ( Jt  ( X  i^i  A ) )  =  ( Jt  ( A  i^i  X ) )
1312a1i 11 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  ( X  i^i  A ) )  =  ( Jt  ( A  i^i  X
) ) )
147, 10, 133eqtr3d 2481 1  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ( Jt  ( A  i^i  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    i^i cin 3324   U.cuni 4088  (class class class)co 6090   ↾t crest 14355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-rest 14357
This theorem is referenced by:  restuni2  18730  cnrest2r  18850  cnrmi  18923  restcnrm  18925  resthauslem  18926  imacmp  18959  fiuncmp  18966  kgeni  19069  ressxms  20059  ptrest  28350
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