Theorem resmptf 5069

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 5067	. 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 2375	. . . 4 |- F/_ y A
4		nfcv 2375	. . . 4 |- F/_ y C
5		nfcsb1v 3161	. . . 4 |- F/_ x [_ y / x ]_ C
6		csbeq1a 3137	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
7	2, 3, 4, 5, 6	cbvmptf 4188	. . 3 |- ( x e. A |-> C ) = ( y e. A |-> [_ y / x ]_ C )
8	7	reseq1i 5015	. 2 |- ( ( x e. A |-> C ) |` B ) = ( ( y e. A |-> [_ y / x ]_ C ) |` B )
9		resmptf.b	. . 3 |- F/_ x B
10		nfcv 2375	. . 3 |- F/_ y B
11	9, 10, 4, 5, 6	cbvmptf 4188	. 2 |- ( x e. B |-> C ) = ( y e. B |-> [_ y / x ]_ C )
12	1, 8, 11	3eqtr4g 2289	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1398   F/_wnfc 2362   [_csb 3128   C_ wss 3201   |-> cmpt 4155   |` cres 4733

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738  df-res 4743

This theorem is referenced by: (None)