Theorem funfvima3 6495

Description: A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)

Assertion

Ref	Expression
funfvima3	|- ( ( Fun F /\ F C_ G ) -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) )

Proof of Theorem funfvima3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvop 6329	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		ssel 3597	. . . . . 6 |- ( F C_ G -> ( <. A , ( F ` A ) >. e. F -> <. A , ( F ` A ) >. e. G ) )
3	1, 2	syl5 34	. . . . 5 |- ( F C_ G -> ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. G ) )
4	3	imp 445	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> <. A , ( F ` A ) >. e. G )
5		sneq 4187	. . . . . . . 8 |- ( x = A -> { x } = { A } )
6	5	imaeq2d 5466	. . . . . . 7 |- ( x = A -> ( G " { x } ) = ( G " { A } ) )
7	6	eleq2d 2687	. . . . . 6 |- ( x = A -> ( ( F ` A ) e. ( G " { x } ) <-> ( F ` A ) e. ( G " { A } ) ) )
8		opeq1 4402	. . . . . . 7 |- ( x = A -> <. x , ( F ` A ) >. = <. A , ( F ` A ) >. )
9	8	eleq1d 2686	. . . . . 6 |- ( x = A -> ( <. x , ( F ` A ) >. e. G <-> <. A , ( F ` A ) >. e. G ) )
10		vex 3203	. . . . . . 7 |- x e. _V
11		fvex 6201	. . . . . . 7 |- ( F ` A ) e. _V
12	10, 11	elimasn 5490	. . . . . 6 |- ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G )
13	7, 9, 12	vtoclbg 3267	. . . . 5 |- ( A e. dom F -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
14	13	ad2antll 765	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
15	4, 14	mpbird 247	. . 3 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( F ` A ) e. ( G " { A } ) )
16	15	exp32 631	. 2 |- ( F C_ G -> ( Fun F -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) ) )
17	16	impcom 446	1 |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   <.cop 4183   dom cdm 5114   "cima 5117   Fun wfun 5882   ` 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-fn 5891  df-fv 5896

This theorem is referenced by:  dfac3  8944