Theorem fvmpts 5596

Description: Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
fvmpts.1	|- F = ( x e. C |-> B )

Assertion

Ref	Expression
fvmpts	|- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)   F( x)   V( x)

Proof of Theorem fvmpts

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3062	. 2 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
2		fvmpts.1	. . 3 |- F = ( x e. C |-> B )
3		nfcv 2319	. . . 4 |- F/_ y B
4		nfcsb1v 3092	. . . 4 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3068	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4100	. . 3 |- ( x e. C |-> B ) = ( y e. C |-> [_ y / x ]_ B )
7	2, 6	eqtri 2198	. 2 |- F = ( y e. C |-> [_ y / x ]_ B )
8	1, 7	fvmptg 5594	1 |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   [_csb 3059   |-> cmpt 4066   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226

This theorem is referenced by:  fvmptd  5599  cc3  7269  sumfct  11384  zsumdc  11394  isumss  11401  fsummulc2  11458  isumshft  11500  prodfct  11597  prodssdc  11599  fprodmul  11601  mulcncflem  14175