Theorem resmptf 4941

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 4939	. 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 2312	. . . 4 |- F/_ y A
4		nfcv 2312	. . . 4 |- F/_ y C
5		nfcsb1v 3082	. . . 4 |- F/_ x [_ y / x ]_ C
6		csbeq1a 3058	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
7	2, 3, 4, 5, 6	cbvmptf 4083	. . 3 |- ( x e. A |-> C ) = ( y e. A |-> [_ y / x ]_ C )
8	7	reseq1i 4887	. 2 |- ( ( x e. A |-> C ) |` B ) = ( ( y e. A |-> [_ y / x ]_ C ) |` B )
9		resmptf.b	. . 3 |- F/_ x B
10		nfcv 2312	. . 3 |- F/_ y B
11	9, 10, 4, 5, 6	cbvmptf 4083	. 2 |- ( x e. B |-> C ) = ( y e. B |-> [_ y / x ]_ C )
12	1, 8, 11	3eqtr4g 2228	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1348   F/_wnfc 2299   [_csb 3049   C_ wss 3121   |-> cmpt 4050   |` cres 4613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by: (None)