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Theorem resmptf 27320
Description: Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
Hypotheses
Ref Expression
resmptf.a  |-  F/_ x A
resmptf.b  |-  F/_ x B
Assertion
Ref Expression
resmptf  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )

Proof of Theorem resmptf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 resmpt 5329 . 2  |-  ( B 
C_  A  ->  (
( y  e.  A  |-> 
[_ y  /  x ]_ C )  |`  B )  =  ( y  e.  B  |->  [_ y  /  x ]_ C ) )
2 resmptf.a . . . 4  |-  F/_ x A
3 nfcv 2629 . . . 4  |-  F/_ y A
4 nfcv 2629 . . . 4  |-  F/_ y C
5 nfcsb1v 3456 . . . 4  |-  F/_ x [_ y  /  x ]_ C
6 csbeq1a 3449 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
72, 3, 4, 5, 6cbvmptf 27317 . . 3  |-  ( x  e.  A  |->  C )  =  ( y  e.  A  |->  [_ y  /  x ]_ C )
87reseq1i 5275 . 2  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( ( y  e.  A  |->  [_ y  /  x ]_ C )  |`  B )
9 resmptf.b . . 3  |-  F/_ x B
10 nfcv 2629 . . 3  |-  F/_ y B
119, 10, 4, 5, 6cbvmptf 27317 . 2  |-  ( x  e.  B  |->  C )  =  ( y  e.  B  |->  [_ y  /  x ]_ C )
121, 8, 113eqtr4g 2533 1  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   F/_wnfc 2615   [_csb 3440    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-sbc 3337  df-csb 3441  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:  esumval  27882  esumel  27883  esumsplit  27888  esumss  27903
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