Theorem resmptf 4775

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 4773	. 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 2229	. . . 4 |- F/_ y A
4		nfcv 2229	. . . 4 |- F/_ y C
5		nfcsb1v 2964	. . . 4 |- F/_ x [_ y / x ]_ C
6		csbeq1a 2942	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
7	2, 3, 4, 5, 6	cbvmptf 3938	. . 3 |- ( x e. A |-> C ) = ( y e. A |-> [_ y / x ]_ C )
8	7	reseq1i 4722	. 2 |- ( ( x e. A |-> C ) |` B ) = ( ( y e. A |-> [_ y / x ]_ C ) |` B )
9		resmptf.b	. . 3 |- F/_ x B
10		nfcv 2229	. . 3 |- F/_ y B
11	9, 10, 4, 5, 6	cbvmptf 3938	. 2 |- ( x e. B |-> C ) = ( y e. B |-> [_ y / x ]_ C )
12	1, 8, 11	3eqtr4g 2146	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1290   F/_wnfc 2216   [_csb 2934   C_ wss 3000   |-> cmpt 3905   |` cres 4454

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-mpt 3907  df-xp 4458  df-rel 4459  df-res 4464

This theorem is referenced by: (None)