Theorem fvmpti 6281

Description: Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fvmptg.1	|- ( x = A -> B = C )
fvmptg.2	|- F = ( x e. D |-> B )

Assertion

Ref	Expression
fvmpti	|- ( A e. D -> ( F ` A ) = ( _I ` C ) )

Distinct variable groups:   x, A   x, C   x, D

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

Proof of Theorem fvmpti

Step	Hyp	Ref	Expression
1		fvmptg.1	. . . 4 |- ( x = A -> B = C )
2		fvmptg.2	. . . 4 |- F = ( x e. D |-> B )
3	1, 2	fvmptg 6280	. . 3 |- ( ( A e. D /\ C e. _V ) -> ( F ` A ) = C )
4		fvi 6255	. . . 4 |- ( C e. _V -> ( _I ` C ) = C )
5	4	adantl 482	. . 3 |- ( ( A e. D /\ C e. _V ) -> ( _I ` C ) = C )
6	3, 5	eqtr4d 2659	. 2 |- ( ( A e. D /\ C e. _V ) -> ( F ` A ) = ( _I ` C ) )
7	1	eleq1d 2686	. . . . . . . 8 |- ( x = A -> ( B e. _V <-> C e. _V ) )
8	2	dmmpt 5630	. . . . . . . 8 |- dom F = { x e. D | B e. _V }
9	7, 8	elrab2 3366	. . . . . . 7 |- ( A e. dom F <-> ( A e. D /\ C e. _V ) )
10	9	baib 944	. . . . . 6 |- ( A e. D -> ( A e. dom F <-> C e. _V ) )
11	10	notbid 308	. . . . 5 |- ( A e. D -> ( -. A e. dom F <-> -. C e. _V ) )
12		ndmfv 6218	. . . . 5 |- ( -. A e. dom F -> ( F ` A ) = (/) )
13	11, 12	syl6bir 244	. . . 4 |- ( A e. D -> ( -. C e. _V -> ( F ` A ) = (/) ) )
14	13	imp 445	. . 3 |- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = (/) )
15		fvprc 6185	. . . 4 |- ( -. C e. _V -> ( _I ` C ) = (/) )
16	15	adantl 482	. . 3 |- ( ( A e. D /\ -. C e. _V ) -> ( _I ` C ) = (/) )
17	14, 16	eqtr4d 2659	. 2 |- ( ( A e. D /\ -. C e. _V ) -> ( F ` A ) = ( _I ` C ) )
18	6, 17	pm2.61dan 832	1 |- ( A e. D -> ( F ` A ) = ( _I ` C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvmpt2i  6290  fvmptex  6294  sumeq2ii  14423  summolem3  14445  fsumf1o  14454  isumshft  14571  prodeq2ii  14643  prodmolem3  14663  fprodf1o  14676