Theorem fvmpts 5564

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 3048	. 2 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
2		fvmpts.1	. . 3 |- F = ( x e. C |-> B )
3		nfcv 2308	. . . 4 |- F/_ y B
4		nfcsb1v 3078	. . . 4 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3054	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4077	. . 3 |- ( x e. C |-> B ) = ( y e. C |-> [_ y / x ]_ B )
7	2, 6	eqtri 2186	. 2 |- F = ( y e. C |-> [_ y / x ]_ B )
8	1, 7	fvmptg 5562	1 |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   [_csb 3045   |-> cmpt 4043   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by:  fvmptd  5567  cc3  7209  sumfct  11315  zsumdc  11325  isumss  11332  fsummulc2  11389  isumshft  11431  prodfct  11528  prodssdc  11530  fprodmul  11532  mulcncflem  13230