Theorem fvmpt2 5275

Description: Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Hypothesis

Ref	Expression
fvmpt2.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fvmpt2	|- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)   F( x)

Proof of Theorem fvmpt2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 2911	. . 3 |- ( y = x -> [_ y / x ]_ B = [_ x / x ]_ B )
2		csbid 2915	. . 3 |- [_ x / x ]_ B = B
3	1, 2	syl6eq 2129	. 2 |- ( y = x -> [_ y / x ]_ B = B )
4		fvmpt2.1	. . 3 |- F = ( x e. A |-> B )
5		nfcv 2219	. . . 4 |- F/_ y B
6		nfcsb1v 2938	. . . 4 |- F/_ x [_ y / x ]_ B
7		csbeq1a 2916	. . . 4 |- ( x = y -> B = [_ y / x ]_ B )
8	5, 6, 7	cbvmpt 3872	. . 3 |- ( x e. A |-> B ) = ( y e. A |-> [_ y / x ]_ B )
9	4, 8	eqtri 2101	. 2 |- F = ( y e. A |-> [_ y / x ]_ B )
10	3, 9	fvmptg 5269	1 |- ( ( x e. A /\ B e. C ) -> ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   [_csb 2908   |-> cmpt 3839   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by:  fvmptssdm  5276  fvmpt2d  5278  fvmptdf  5279  mpteqb  5282  fvmptt  5283  fvmptf  5284  ralrnmpt  5330  rexrnmpt  5331  fmptco  5351  f1mpt  5431  offval2  5746  ofrfval2  5747  dom2lem  6275