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Theorem resmpt3 4774
Description: Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
Assertion
Ref Expression
resmpt3  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B
)  |->  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem resmpt3
StepHypRef Expression
1 resres 4738 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )
2 ssid 3045 . . . 4  |-  A  C_  A
3 resmpt 4773 . . . 4  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
42, 3ax-mp 7 . . 3  |-  ( ( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C )
54reseq1i 4722 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  B )
6 inss1 3221 . . 3  |-  ( A  i^i  B )  C_  A
7 resmpt 4773 . . 3  |-  ( ( A  i^i  B ) 
C_  A  ->  (
( x  e.  A  |->  C )  |`  ( A  i^i  B ) )  =  ( x  e.  ( A  i^i  B
)  |->  C ) )
86, 7ax-mp 7 . 2  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )  =  ( x  e.  ( A  i^i  B ) 
|->  C )
91, 5, 83eqtr3i 2117 1  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B
)  |->  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1290    i^i cin 2999    C_ wss 3000    |-> cmpt 3905    |` cres 4454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-mpt 3907  df-xp 4458  df-rel 4459  df-res 4464
This theorem is referenced by:  offres  5920
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