Theorem fvrn 2888

Description: A function's value belongs to its range.

Assertion

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

Proof of Theorem fvrn

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 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 2832	. . . . 5 |- (x = A -> (F` x) = (F` A))
4	3	eleq1d 1155	. . . 4 |- (x = A -> ((F` x) e. ran F <-> (F` A) e. ran F))
5	2, 4	imbi12d 474	. . 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		funopfv 2886	. . . . 5 |- ((Fun F /\ x e. dom F) -> <.x, (F` x)>. e. F)
7		visset 1350	. . . . . 6 |- x e. V
8		opeq1 1876	. . . . . . 7 |- (y = x -> <.y, (F` x)>. = <.x, (F` x)>.)
9	8	eleq1d 1155	. . . . . 6 |- (y = x -> (<.y, (F` x)>. e. F <-> <.x, (F` x)>. e. F))
10	7, 9	cla4ev 1401	. . . . 5 |- (<.x, (F` x)>. e. F -> E.y<.y, (F` x)>. e. F)
11	6, 10	syl 12	. . . 4 |- ((Fun F /\ x e. dom F) -> E.y<.y, (F` x)>. e. F)
12		fvex 2838	. . . . 5 |- (F` x) e. V
13	12	elrn 2562	. . . 4 |- ((F` x) e. ran F <-> E.y<.y, (F` x)>. e. F)
14	11, 13	sylibr 175	. . 3 |- ((Fun F /\ x e. dom F) -> (F` x) e. ran F)
15	5, 14	vtoclg 1383	. 2 |- (A e. dom F -> ((Fun F /\ A e. dom F) -> (F` A) e. ran F))
16	15	anabsi7 379	1 |- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  <.cop 1810  dom cdm 2410  ran crn 2411  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  fnfvrn 2889  funfvima 2904  tz7.48-2 2995

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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