Theorem fvelrn 3803

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 1531	. . . . 5 |- (x = A -> (x e. dom F <-> A e. dom F))
2	1	anbi2d 615	. . . 4 |- (x = A -> ((Fun F /\ x e. dom F) <-> (Fun F /\ A e. dom F)))
3		fveq2 3715	. . . . 5 |- (x = A -> (F` x) = (F` A))
4	3	eleq1d 1537	. . . 4 |- (x = A -> ((F` x) e. ran F <-> (F` A) e. ran F))
5	2, 4	imbi12d 625	. . 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 3794	. . . . 5 |- ((Fun F /\ x e. dom F) -> <.x, (F` x)>. e. F)
7		visset 1809	. . . . . 6 |- x e. V
8		opeq1 2483	. . . . . . 7 |- (y = x -> <.y, (F` x)>. = <.x, (F` x)>.)
9	8	eleq1d 1537	. . . . . 6 |- (y = x -> (<.y, (F` x)>. e. F <-> <.x, (F` x)>. e. F))
10	7, 9	cla4ev 1865	. . . . 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 3723	. . . . 5 |- (F` x) e. V
13	12	elrn2 3343	. . . 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 1843	. 2 |- (A e. dom F -> ((Fun F /\ A e. dom F) -> (F` A) e. ran F))
16	15	anabsi7 497	1 |- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  <.cop 2407  dom cdm 3165  ran crn 3166  Fun wfun 3171  ` cfv 3177

This theorem is referenced by:  fnfvelrn 3804  funfvima 3843  elunirnALT 3860  tz7.48-2 3948  fnoprvalrn2 10402  rdmob 10561  rcmob 10562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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