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Theorem restin 12359
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 4361 . . . . 5  |-  ( J  e.  V  ->  U. J  e.  _V )
31, 2eqeltrid 2226 . . . 4  |-  ( J  e.  V  ->  X  e.  _V )
43adantr 274 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  X  e.  _V )
5 restco 12357 . . . 4  |-  ( ( J  e.  V  /\  X  e.  _V  /\  A  e.  W )  ->  (
( Jt  X )t  A )  =  ( Jt  ( X  i^i  A
) ) )
653com23 1187 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  X  e.  _V )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
74, 6mpd3an3 1316 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  ( X  i^i  A ) ) )
81restid 12145 . . . 4  |-  ( J  e.  V  ->  ( Jt  X )  =  J )
98adantr 274 . . 3  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  X )  =  J )
109oveq1d 5789 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( ( Jt  X )t  A )  =  ( Jt  A ) )
11 incom 3268 . . . 4  |-  ( X  i^i  A )  =  ( A  i^i  X
)
1211oveq2i 5785 . . 3  |-  ( Jt  ( X  i^i  A ) )  =  ( Jt  ( A  i^i  X ) )
1312a1i 9 . 2  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  ( X  i^i  A ) )  =  ( Jt  ( A  i^i  X
) ) )
147, 10, 133eqtr3d 2180 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 103    = wceq 1331    e. wcel 1480   _Vcvv 2686    i^i cin 3070   U.cuni 3736  (class class class)co 5774   ↾t crest 12134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-rest 12136
This theorem is referenced by:  restuni2  12360  cnrest2r  12420
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