Theorem funfvima3 5700

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 5579	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		ssel 3122	. . . . . 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 123	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> <. A , ( F ` A ) >. e. G )
5		simpr 109	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> A e. dom F )
6		sneq 3571	. . . . . . . . . 10 |- ( x = A -> { x } = { A } )
7	6	imaeq2d 4928	. . . . . . . . 9 |- ( x = A -> ( G " { x } ) = ( G " { A } ) )
8	7	eleq2d 2227	. . . . . . . 8 |- ( x = A -> ( ( F ` A ) e. ( G " { x } ) <-> ( F ` A ) e. ( G " { A } ) ) )
9		opeq1 3741	. . . . . . . . 9 |- ( x = A -> <. x , ( F ` A ) >. = <. A , ( F ` A ) >. )
10	9	eleq1d 2226	. . . . . . . 8 |- ( x = A -> ( <. x , ( F ` A ) >. e. G <-> <. A , ( F ` A ) >. e. G ) )
11	8, 10	bibi12d 234	. . . . . . 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 275	. . . . . 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 2715	. . . . . . 7 |- x e. _V
14		funfvex 5485	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
15		elimasng 4954	. . . . . . 7 |- ( ( x e. _V /\ ( F ` A ) e. _V ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
16	13, 14, 15	sylancr 411	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
17	5, 12, 16	vtocld 2764	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
18	17	adantl 275	. . . 4 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( ( F ` A ) e. ( G " { A } ) <-> <. A , ( F ` A ) >. e. G ) )
19	4, 18	mpbird 166	. . 3 |- ( ( F C_ G /\ ( Fun F /\ A e. dom F ) ) -> ( F ` A ) e. ( G " { A } ) )
20	19	exp32 363	. 2 |- ( F C_ G -> ( Fun F -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) ) )
21	20	impcom 124	1 |- ( ( Fun F /\ F C_ G ) -> ( A e. dom F -> ( F ` A ) e. ( G " { A } ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   _Vcvv 2712   C_ wss 3102   {csn 3560   <.cop 3563   dom cdm 4586   "cima 4589   Fun wfun 5164   ` cfv 5170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178

This theorem is referenced by: (None)