Theorem fvelrn 5830

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 2301	. . . . 5 |- ( x = A -> ( x e. dom F <-> A e. dom F ) )
2	1	anbi2d 468	. . . 4 |- ( x = A -> ( ( Fun F /\ x e. dom F ) <-> ( Fun F /\ A e. dom F ) ) )
3		fveq2 5690	. . . . 5 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eleq1d 2307	. . . 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 5812	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
7		vex 2824	. . . . . 6 |- x e. _V
8		opeq1 3899	. . . . . . 7 |- ( y = x -> <. y , ( F ` x ) >. = <. x , ( F ` x ) >. )
9	8	eleq1d 2307	. . . . . 6 |- ( y = x -> ( <. y , ( F ` x ) >. e. F <-> <. x , ( F ` x ) >. e. F ) )
10	7, 9	spcev 2920	. . . . 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 5707	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
13		elrn2g 4965	. . . . 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 2883	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) )
17	16	anabsi7 587	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   dom cdm 4769   ran crn 4770   Fun wfun 5366   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by:  fnfvelrn  5831  eldmrexrn  5840  funfvima  5940  elunirn  5962  frecuzrdgdomlem  10832  frecuzrdgsuctlem  10838  iedgedgg  16216  usgredg3  16369  ushgredgedg  16381  ushgredgedgloop  16383  subgruhgredgdm  16425  edginwlkd  16510  iedginwlk  16512