Theorem funfvima3 5651

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 5532	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> <. A , ( F ` A ) >. e. F )
2		ssel 3091	. . . . . 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 3538	. . . . . . . . . 10 |- ( x = A -> { x } = { A } )
7	6	imaeq2d 4881	. . . . . . . . 9 |- ( x = A -> ( G " { x } ) = ( G " { A } ) )
8	7	eleq2d 2209	. . . . . . . 8 |- ( x = A -> ( ( F ` A ) e. ( G " { x } ) <-> ( F ` A ) e. ( G " { A } ) ) )
9		opeq1 3705	. . . . . . . . 9 |- ( x = A -> <. x , ( F ` A ) >. = <. A , ( F ` A ) >. )
10	9	eleq1d 2208	. . . . . . . 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 2689	. . . . . . 7 |- x e. _V
14		funfvex 5438	. . . . . . 7 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. _V )
15		elimasng 4907	. . . . . . 7 |- ( ( x e. _V /\ ( F ` A ) e. _V ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
16	13, 14, 15	sylancr 410	. . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) e. ( G " { x } ) <-> <. x , ( F ` A ) >. e. G ) )
17	5, 12, 16	vtocld 2738	. . . . 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 362	. 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 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   {csn 3527   <.cop 3530   dom cdm 4539   "cima 4542   Fun wfun 5117   ` cfv 5123

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by: (None)