Theorem fvelrn 5689

Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Assertion

Ref	Expression
fvelrn	|- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )

Proof of Theorem fvelrn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2256	. . . . 5 |- ( x = A -> ( x e. dom F <-> A e. dom F ) )
2	1	anbi2d 464	. . . 4 |- ( x = A -> ( ( Fun F /\ x e. dom F ) <-> ( Fun F /\ A e. dom F ) ) )
3		fveq2 5554	. . . . 5 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eleq1d 2262	. . . 4 |- ( x = A -> ( ( F ` x ) e. ran F <-> ( F ` A ) e. ran F ) )
5	2, 4	imbi12d 234	. . 3 |- ( x = A -> ( ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. ran F ) <-> ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) ) )
6		funfvop 5670	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
7		vex 2763	. . . . . 6 |- x e. _V
8		opeq1 3804	. . . . . . 7 |- ( y = x -> <. y , ( F ` x ) >. = <. x , ( F ` x ) >. )
9	8	eleq1d 2262	. . . . . 6 |- ( y = x -> ( <. y , ( F ` x ) >. e. F <-> <. x , ( F ` x ) >. e. F ) )
10	7, 9	spcev 2855	. . . . 5 |- ( <. x , ( F ` x ) >. e. F -> E. y <. y , ( F ` x ) >. e. F )
11	6, 10	syl 14	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> E. y <. y , ( F ` x ) >. e. F )
12		funfvex 5571	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
13		elrn2g 4852	. . . . 5 |- ( ( F ` x ) e. _V -> ( ( F ` x ) e. ran F <-> E. y <. y , ( F ` x ) >. e. F ) )
14	12, 13	syl 14	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. ran F <-> E. y <. y , ( F ` x ) >. e. F ) )
15	11, 14	mpbird 167	. . 3 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. ran F )
16	5, 15	vtoclg 2820	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) )
17	16	anabsi7 581	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   <.cop 3621   dom cdm 4659   ran crn 4660   Fun wfun 5248   ` cfv 5254

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  fnfvelrn  5690  eldmrexrn  5699  funfvima  5790  elunirn  5809  frecuzrdgdomlem  10488  frecuzrdgsuctlem  10494  gsumpropd2  12976