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Theorem resmpt3 4681
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 4646 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )
2 ssid 3019 . . . 4  |-  A  C_  A
3 resmpt 4680 . . . 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 4630 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  B )
6 inss1 3187 . . 3  |-  ( A  i^i  B )  C_  A
7 resmpt 4680 . . 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 2110 1  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B
)  |->  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    i^i cin 2973    C_ wss 2974    |-> cmpt 3841    |` cres 4367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842  df-mpt 3843  df-xp 4371  df-rel 4372  df-res 4377
This theorem is referenced by:  offres  5787
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