Theorem funfvima3 5873

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 5747	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		ssel 3218	. . . . . 6 |- ( F C_ G -> ( <. A , ( F ` A ) >. e. F -> <. A , ( F ` A ) >. e. G ) )
3	1, 2	syl5 32	. . . . 5 |- ( F C_ G -> ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. G ) )
4	3	imp 124	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> <. A , ( F ` A ) >. e. G )
5		simpr 110	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> A e. dom F )
6		sneq 3677	. . . . . . . . . 10 |- ( x = A -> { x } = { A } )
7	6	imaeq2d 5068	. . . . . . . . 9 |- ( x = A -> ( G " { x } ) = ( G " { A } ) )
8	7	eleq2d 2299	. . . . . . . 8 |- ( x = A -> ( ( F ` A ) e. ( G " { x } ) <-> ( F ` A ) e. ( G " { A } ) ) )
9		opeq1 3857	. . . . . . . . 9 |- ( x = A -> <. x , ( F ` A ) >. = <. A , ( F ` A ) >. )
10	9	eleq1d 2298	. . . . . . . 8 |- ( x = A -> ( <. x , ( F ` A ) >. e. G <-> <. A , ( F ` A ) >. e. G ) )
11	8, 10	bibi12d 235	. . . . . . 7 |- ( x = A -> ( ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) <-> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) ) )
12	11	adantl 277	. . . . . 6 |- ( ( ( Fun F /\ A e. dom F ) /\ x = A ) -> ( ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) <-> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) ) )
13		vex 2802	. . . . . . 7 |- x e. _V
14		funfvex 5644	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
15		elimasng 5096	. . . . . . 7 |- ( ( x e. _V /\ ( F ` A ) e. _V ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
16	13, 14, 15	sylancr 414	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
17	5, 12, 16	vtocld 2853	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
18	17	adantl 277	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
19	4, 18	mpbird 167	. . 3 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( F ` A ) e. ( G " { A } ) )
20	19	exp32 365	. 2 |- ( F C_ G -> ( Fun F -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) ) )
21	20	impcom 125	1 |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {csn 3666   <.cop 3669   dom cdm 4719   "cima 4722   Fun wfun 5312   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by: (None)