Theorem resmptf 5008

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.

Step	Hyp	Ref	Expression
1		resmpt 5006	. 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 2347	. . . 4 |- F/_ y A
4		nfcv 2347	. . . 4 |- F/_ y C
5		nfcsb1v 3125	. . . 4 |- F/_ x [_ y / x ]_ C
6		csbeq1a 3101	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
7	2, 3, 4, 5, 6	cbvmptf 4137	. . 3 |- ( x e. A |-> C ) = ( y e. A |-> [_ y / x ]_ C )
8	7	reseq1i 4954	. 2 |- ( ( x e. A |-> C ) |` B ) = ( ( y e. A |-> [_ y / x ]_ C ) |` B )
9		resmptf.b	. . 3 |- F/_ x B
10		nfcv 2347	. . 3 |- F/_ y B
11	9, 10, 4, 5, 6	cbvmptf 4137	. 2 |- ( x e. B |-> C ) = ( y e. B |-> [_ y / x ]_ C )
12	1, 8, 11	3eqtr4g 2262	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1372   F/_wnfc 2334   [_csb 3092   C_ wss 3165   |-> cmpt 4104   |` cres 4676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-mpt 4106  df-xp 4680  df-rel 4681  df-res 4686

This theorem is referenced by: (None)