Theorem funfvima3 5796

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 5674	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		ssel 3177	. . . . . 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 3633	. . . . . . . . . 10 |- ( x = A -> { x } = { A } )
7	6	imaeq2d 5009	. . . . . . . . 9 |- ( x = A -> ( G " { x } ) = ( G " { A } ) )
8	7	eleq2d 2266	. . . . . . . 8 |- ( x = A -> ( ( F ` A ) e. ( G " { x } ) <-> ( F ` A ) e. ( G " { A } ) ) )
9		opeq1 3808	. . . . . . . . 9 |- ( x = A -> <. x , ( F ` A ) >. = <. A , ( F ` A ) >. )
10	9	eleq1d 2265	. . . . . . . 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 2766	. . . . . . 7 |- x e. _V
14		funfvex 5575	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
15		elimasng 5037	. . . . . . 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 2816	. . . . 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 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   {csn 3622   <.cop 3625   dom cdm 4663   "cima 4666   Fun wfun 5252   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by: (None)