Theorem fvelrn 5786

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 2294	. . . . 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 5648	. . . . 5 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eleq1d 2300	. . . 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 5768	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> <. x , ( F ` x ) >. e. F )
7		vex 2806	. . . . . 6 |- x e. _V
8		opeq1 3867	. . . . . . 7 |- ( y = x -> <. y , ( F ` x ) >. = <. x , ( F ` x ) >. )
9	8	eleq1d 2300	. . . . . 6 |- ( y = x -> ( <. y , ( F ` x ) >. e. F <-> <. x , ( F ` x ) >. e. F ) )
10	7, 9	spcev 2902	. . . . 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 5665	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
13		elrn2g 4926	. . . . 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 2865	. 2 |- ( A e. dom F -> ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F ) )
17	16	anabsi7 583	1 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   <.cop 3676   dom cdm 4731   ran crn 4732   Fun wfun 5327   ` cfv 5333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  fnfvelrn  5787  eldmrexrn  5796  funfvima  5896  elunirn  5917  frecuzrdgdomlem  10742  frecuzrdgsuctlem  10748  gsumpropd2  13556  iedgedgg  16002  usgredg3  16155  ushgredgedg  16167  ushgredgedgloop  16169  subgruhgredgdm  16211  edginwlkd  16296  iedginwlk  16298