Theorem funfvima3 5392

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 5279	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		ssel 2939	. . . . . 6 |- ( F C_ G -> ( <. A , ( F ` A ) >. e. F -> <. A , ( F ` A ) >. e. G ) )
3	1, 2	syl5 28	. . . . 5 |- ( F C_ G -> ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. G ) )
4	3	imp 115	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> <. A , ( F ` A ) >. e. G )
5		simpr 103	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> A e. dom F )
6		sneq 3386	. . . . . . . . . 10 |- ( x = A -> { x } = { A } )
7	6	imaeq2d 4668	. . . . . . . . 9 |- ( x = A -> ( G " { x } ) = ( G " { A } ) )
8	7	eleq2d 2107	. . . . . . . 8 |- ( x = A -> ( ( F ` A ) e. ( G " { x } ) <-> ( F ` A ) e. ( G " { A } ) ) )
9		opeq1 3549	. . . . . . . . 9 |- ( x = A -> <. x , ( F ` A ) >. = <. A , ( F ` A ) >. )
10	9	eleq1d 2106	. . . . . . . 8 |- ( x = A -> ( <. x , ( F ` A ) >. e. G <-> <. A , ( F ` A ) >. e. G ) )
11	8, 10	bibi12d 224	. . . . . . 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 262	. . . . . 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 2560	. . . . . . 7 |- x e. _V
14		funfvex 5192	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
15		elimasng 4693	. . . . . . 7 |- ( ( x e. _V /\ ( F ` A ) e. _V ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
16	13, 14, 15	sylancr 393	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
17	5, 12, 16	vtocld 2606	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
18	17	adantl 262	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
19	4, 18	mpbird 156	. . 3 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( F ` A ) e. ( G " { A } ) )
20	19	exp32 347	. 2 |- ( F C_ G -> ( Fun F -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) ) )
21	20	impcom 116	1 |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   {csn 3375   <.cop 3378   dom cdm 4345   "cima 4348   Fun wfun 4896   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)