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Theorem resmpt3 4838
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 4801 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  ( A  i^i  B ) )
2 ssid 3087 . . . 4  |-  A  C_  A
3 resmpt 4837 . . . 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 4785 . 2  |-  ( ( ( x  e.  A  |->  C )  |`  A )  |`  B )  =  ( ( x  e.  A  |->  C )  |`  B )
6 inss1 3266 . . 3  |-  ( A  i^i  B )  C_  A
7 resmpt 4837 . . 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 2146 1  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B
)  |->  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    i^i cin 3040    C_ wss 3041    |-> cmpt 3959    |` cres 4511
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516  df-res 4521
This theorem is referenced by:  offres  6001
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