Theorem resmptf 5451

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 5449	. 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 2764	. . . 4 |- F/_ y A
4		nfcv 2764	. . . 4 |- F/_ y C
5		nfcsb1v 3549	. . . 4 |- F/_ x [_ y / x ]_ C
6		csbeq1a 3542	. . . 4 |- ( x = y -> C = [_ y / x ]_ C )
7	2, 3, 4, 5, 6	cbvmptf 4748	. . 3 |- ( x e. A |-> C ) = ( y e. A |-> [_ y / x ]_ C )
8	7	reseq1i 5392	. 2 |- ( ( x e. A |-> C ) |` B ) = ( ( y e. A |-> [_ y / x ]_ C ) |` B )
9		resmptf.b	. . 3 |- F/_ x B
10		nfcv 2764	. . 3 |- F/_ y B
11	9, 10, 4, 5, 6	cbvmptf 4748	. 2 |- ( x e. B |-> C ) = ( y e. B |-> [_ y / x ]_ C )
12	1, 8, 11	3eqtr4g 2681	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1483   F/_wnfc 2751   [_csb 3533   C_ wss 3574   |-> cmpt 4729   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  esumval  30108  esumel  30109  esumsplit  30115  esumss  30134  limsupequzmpt2  39950  liminfequzmpt2  40023