Theorem chfnrn 4775

Description: The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain.

Assertion

Ref	Expression
chfnrn	|- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F C_ U.A)

Distinct variable groups:   x,A   x,F

Proof of Theorem chfnrn

Step	Hyp	Ref	Expression
1		fvelrnb 4719	. . . . 5 |- (F Fn A -> (y e. ran F <-> E.x e. A (F` x) = y))
2	1	biimpd 170	. . . 4 |- (F Fn A -> (y e. ran F -> E.x e. A (F` x) = y))
3		hbra1 2147	. . . . 5 |- (A.x e. A (F` x) e. x -> A.xA.x e. A (F` x) e. x)
4		ra4 2155	. . . . . 6 |- (A.x e. A (F` x) e. x -> (x e. A -> (F` x) e. x))
5		eleq1 1957	. . . . . . 7 |- ((F` x) = y -> ((F` x) e. x <-> y e. x))
6	5	biimpcd 172	. . . . . 6 |- ((F` x) e. x -> ((F` x) = y -> y e. x))
7	4, 6	syl6 25	. . . . 5 |- (A.x e. A (F` x) e. x -> (x e. A -> ((F` x) = y -> y e. x)))
8	3, 7	reximdai 2199	. . . 4 |- (A.x e. A (F` x) e. x -> (E.x e. A (F` x) = y -> E.x e. A y e. x))
9	2, 8	sylan9 517	. . 3 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> (y e. ran F -> E.x e. A y e. x))
10		eluni2 3181	. . 3 |- (y e. U.A <-> E.x e. A y e. x)
11	9, 10	syl6ibr 230	. 2 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> (y e. ran F -> y e. U.A))
12	11	ssrdv 2622	1 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F C_ U.A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177  ran crn 3987   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  ac5b 5915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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