Theorem fvmpts 5574

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 3052	. 2 |- ( y = A -> [_ y / x ]_ B = [_ A / x ]_ B )
2		fvmpts.1	. . 3 |- F = ( x e. C |-> B )
3		nfcv 2312	. . . 4 |- F/_ y B
4		nfcsb1v 3082	. . . 4 |- F/_ x [_ y / x ]_ B
5		csbeq1a 3058	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
6	3, 4, 5	cbvmpt 4084	. . 3 |- ( x e. C |-> B ) = ( y e. C |-> [_ y / x ]_ B )
7	2, 6	eqtri 2191	. 2 |- F = ( y e. C |-> [_ y / x ]_ B )
8	1, 7	fvmptg 5572	1 |- ( ( A e. C /\ [_ A / x ]_ B e. V ) -> ( F ` A ) = [_ A / x ]_ B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   [_csb 3049   |-> cmpt 4050   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206

This theorem is referenced by:  fvmptd  5577  cc3  7230  sumfct  11337  zsumdc  11347  isumss  11354  fsummulc2  11411  isumshft  11453  prodfct  11550  prodssdc  11552  fprodmul  11554  mulcncflem  13384