Theorem fvmpt2f 6283

Description: Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)

Hypothesis

Ref	Expression
fvmpt2f.0	|- F/_ x A

Assertion

Ref	Expression
fvmpt2f	|- ( ( x e. A /\ B e. C ) -> ( ( x e. A |-> B ) ` x ) = B )

Proof of Theorem fvmpt2f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . 3 |- ( y = x -> [_ y / x ]_ B = [_ x / x ]_ B )
2		csbid 3541	. . 3 |- [_ x / x ]_ B = B
3	1, 2	syl6eq 2672	. 2 |- ( y = x -> [_ y / x ]_ B = B )
4		fvmpt2f.0	. . 3 |- F/_ x A
5		nfcv 2764	. . 3 |- F/_ y A
6		nfcv 2764	. . 3 |- F/_ y B
7		nfcsb1v 3549	. . 3 |- F/_ x [_ y / x ]_ B
8		csbeq1a 3542	. . 3 |- ( x = y -> B = [_ y / x ]_ B )
9	4, 5, 6, 7, 8	cbvmptf 4748	. 2 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
10	3, 9	fvmptg 6280	1 |- ( ( x e. A /\ B e. C ) -> ( ( x e. A |-> B ) ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   [_csb 3533   |-> cmpt 4729   ` cfv 5888

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  offval2f  6909  fmptcof2  29457  funcnvmptOLD  29467  funcnvmpt  29468  esumc  30113