Theorem fvelrn 3818

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

Assertion

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

Proof of Theorem fvelrn

Step	Hyp	Ref	Expression
1		eleq1 1537	. . . . 5 |- (x = A -> (x e. dom F <-> A e. dom F))
2	1	anbi2d 618	. . . 4 |- (x = A -> ((Fun F /\ x e. dom F) <-> (Fun F /\ A e. dom F)))
3		fveq2 3730	. . . . 5 |- (x = A -> (F` x) = (F` A))
4	3	eleq1d 1543	. . . 4 |- (x = A -> ((F` x) e. ran F <-> (F` A) e. ran F))
5	2, 4	imbi12d 628	. . 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 3809	. . . . 5 |- ((Fun F /\ x e. dom F) -> <.x, (F` x)>. e. F)
7		visset 1816	. . . . . 6 |- x e. V
8		opeq1 2491	. . . . . . 7 |- (y = x -> <.y, (F` x)>. = <.x, (F` x)>.)
9	8	eleq1d 1543	. . . . . 6 |- (y = x -> (<.y, (F` x)>. e. F <-> <.x, (F` x)>. e. F))
10	7, 9	cla4ev 1872	. . . . 5 |- (<.x, (F` x)>. e. F -> E.y<.y, (F` x)>. e. F)
11	6, 10	syl 10	. . . 4 |- ((Fun F /\ x e. dom F) -> E.y<.y, (F` x)>. e. F)
12		fvex 3738	. . . . 5 |- (F` x) e. V
13	12	elrn2 3355	. . . 4 |- ((F` x) e. ran F <-> E.y<.y, (F` x)>. e. F)
14	11, 13	sylibr 200	. . 3 |- ((Fun F /\ x e. dom F) -> (F` x) e. ran F)
15	5, 14	vtoclg 1850	. 2 |- (A e. dom F -> ((Fun F /\ A e. dom F) -> (F` A) e. ran F))
16	15	anabsi7 499	1 |- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  <.cop 2415  dom cdm 3176  ran crn 3177  Fun wfun 3182  ` cfv 3188

This theorem is referenced by:  fnfvelrn 3819  funfvima 3858  elunirnALT 3875  tz7.48-2 3963  fnoprvalrn2 10460  rdmob 10652  rcmob 10653

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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