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Theorem resmptf 25989
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 5171 . 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 2589 . . . 4  |-  F/_ y A
4 nfcv 2589 . . . 4  |-  F/_ y C
5 nfcsb1v 3319 . . . 4  |-  F/_ x [_ y  /  x ]_ C
6 csbeq1a 3312 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
72, 3, 4, 5, 6cbvmptf 25986 . . 3  |-  ( x  e.  A  |->  C )  =  ( y  e.  A  |->  [_ y  /  x ]_ C )
87reseq1i 5121 . 2  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( ( y  e.  A  |->  [_ y  /  x ]_ C )  |`  B )
9 resmptf.b . . 3  |-  F/_ x B
10 nfcv 2589 . . 3  |-  F/_ y B
119, 10, 4, 5, 6cbvmptf 25986 . 2  |-  ( x  e.  B  |->  C )  =  ( y  e.  B  |->  [_ y  /  x ]_ C )
121, 8, 113eqtr4g 2500 1  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   F/_wnfc 2575   [_csb 3303    C_ wss 3343    e. cmpt 4365    |` cres 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-opab 4366  df-mpt 4367  df-xp 4861  df-rel 4862  df-res 4867
This theorem is referenced by:  esumval  26515  esumel  26516  esumsplit  26521  esumss  26536
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