Theorem fvmpts 5720

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 3128	. 2 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
2		fvmpts.1	. . 3 |- F = ( x e. C |-> B )
3		nfcv 2372	. . . 4 |- F/_ y B
4		nfcsb1v 3158	. . . 4 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3134	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4182	. . 3 |- ( x e. C |-> B ) = ( y e. C |-> [_ y / x ]_ B )
7	2, 6	eqtri 2250	. 2 |- F = ( y e. C |-> [_ y / x ]_ B )
8	1, 7	fvmptg 5718	1 |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   [_csb 3125   |-> cmpt 4148   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332

This theorem is referenced by:  fvmptd  5723  cc3  7477  sumfct  11925  zsumdc  11935  isumss  11942  fsummulc2  11999  isumshft  12041  prodfct  12138  prodssdc  12140  fprodmul  12142  gsumfzfsumlemm  14591  mulcncflem  15321