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Theorem restin 19794
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 6596 . . . . 5  |-  ( J  e.  V  ->  U. J  e.  _V )
31, 2syl5eqel 2549 . . . 4  |-  ( J  e.  V  ->  X  e.  _V )
43adantr 465 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  X  e.  _V )
5 restco 19792 . . . 4  |-  ( ( J  e.  V  /\  X  e.  _V  /\  A  e.  W )  ->  (
( Jt  X )t  A )  =  ( Jt  ( X  i^i  A
) ) )
653com23 1202 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  X  e.  _V )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
74, 6mpd3an3 1325 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
81restid 14851 . . . 4  |-  ( J  e.  V  ->  ( Jt  X )  =  J )
98adantr 465 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  X )  =  J )
109oveq1d 6311 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  A ) )
11 incom 3687 . . . 4  |-  ( X  i^i  A )  =  ( A  i^i  X
)
1211oveq2i 6307 . . 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 2506 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 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470   U.cuni 4251  (class class class)co 6296   ↾t crest 14838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-rest 14840
This theorem is referenced by:  restuni2  19795  cnrest2r  19915  cnrmi  19988  restcnrm  19990  resthauslem  19991  imacmp  20024  fiuncmp  20031  kgeni  20164  ressxms  21154  ptrest  30232
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