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Theorem resmpt3 5134
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 5101 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )
2 ssid 3312 . . . 4  |-  A  C_  A
3 resmpt 5133 . . . 4  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
42, 3ax-mp 8 . . 3  |-  ( ( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C )
54reseq1i 5084 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  B )
6 inss1 3506 . . 3  |-  ( A  i^i  B )  C_  A
7 resmpt 5133 . . 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 8 . 2  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )  =  ( x  e.  ( A  i^i  B ) 
|->  C )
91, 5, 83eqtr3i 2417 1  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B
)  |->  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    i^i cin 3264    C_ wss 3265    e. cmpt 4209    |` cres 4822
This theorem is referenced by:  offres  6260  lo1resb  12287  o1resb  12289  measinb2  24373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-opab 4210  df-mpt 4211  df-xp 4826  df-rel 4827  df-res 4832
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