Theorem fvmpts 5733

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 3131	. 2 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
2		fvmpts.1	. . 3 |- F = ( x e. C |-> B )
3		nfcv 2375	. . . 4 |- F/_ y B
4		nfcsb1v 3161	. . . 4 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3137	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4189	. . 3 |- ( x e. C |-> B ) = ( y e. C |-> [_ y / x ]_ B )
7	2, 6	eqtri 2252	. 2 |- F = ( y e. C |-> [_ y / x ]_ B )
8	1, 7	fvmptg 5731	1 |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   [_csb 3128   |-> cmpt 4155   ` cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341

This theorem is referenced by:  fvmptd  5736  cc3  7547  sumfct  12014  zsumdc  12025  isumss  12032  fsummulc2  12089  isumshft  12131  prodfct  12228  prodssdc  12230  fprodmul  12232  gsumfzfsumlemm  14683  mulcncflem  15418