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Theorem resmpt3 5330
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 5292 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )
2 ssid 3528 . . . 4  |-  A  C_  A
3 resmpt 5329 . . . 4  |-  ( A 
C_  A  ->  (
( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C ) )
42, 3ax-mp 5 . . 3  |-  ( ( x  e.  A  |->  C )  |`  A )  =  ( x  e.  A  |->  C )
54reseq1i 5275 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  B )
6 inss1 3723 . . 3  |-  ( A  i^i  B )  C_  A
7 resmpt 5329 . . 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 5 . 2  |-  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )  =  ( x  e.  ( A  i^i  B ) 
|->  C )
91, 5, 83eqtr3i 2504 1  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B
)  |->  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    i^i cin 3480    C_ wss 3481    |-> cmpt 4511    |` cres 5007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-mpt 4513  df-xp 5011  df-rel 5012  df-res 5017
This theorem is referenced by:  offres  6790  lo1resb  13366  o1resb  13368  measinb2  28026  eulerpartgbij  28143
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