Theorem fvelrn 5559

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 2203	. . . . 5 |- ( x = A -> ( x e. dom F <-> A e. dom F ) )
2	1	anbi2d 460	. . . 4 |- ( x = A -> ( ( Fun F /\ x e. dom F ) <-> ( Fun F /\ A e. dom F ) ) )
3		fveq2 5429	. . . . 5 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eleq1d 2209	. . . 4 |- ( x = A -> ( ( F ` x ) e. ran F <-> ( F ` A ) e. ran F ) )
5	2, 4	imbi12d 233	. . 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 5540	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
7		vex 2692	. . . . . 6 |- x e. _V
8		opeq1 3713	. . . . . . 7 |- ( y = x -> <. y , ( F ` x ) >. = <. x , ( F ` x ) >. )
9	8	eleq1d 2209	. . . . . 6 |- ( y = x -> ( <. y , ( F ` x ) >. e. F <-> <. x , ( F ` x ) >. e. F ) )
10	7, 9	spcev 2784	. . . . 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 5446	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
13		elrn2g 4737	. . . . 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 166	. . 3 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. ran F )
16	5, 15	vtoclg 2749	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) )
17	16	anabsi7 571	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   <.cop 3535   dom cdm 4547   ran crn 4548   Fun wfun 5125   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  fnfvelrn  5560  eldmrexrn  5569  funfvima  5657  elunirn  5675  frecuzrdgdomlem  10221  frecuzrdgsuctlem  10227