Theorem chfnrn 5524

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. (Contributed by NM, 31-Aug-1999.)

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

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvelrnb 5462	. . . . 5 |- ( F Fn A -> ( y e. ran F <-> E. x e. A ( F ` x ) = y ) )
2	1	biimpd 143	. . . 4 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F ` x ) = y ) )
3		eleq1 2200	. . . . . . 7 |- ( ( F ` x ) = y -> ( ( F ` x ) e. x <-> y e. x ) )
4	3	biimpcd 158	. . . . . 6 |- ( ( F ` x ) e. x -> ( ( F ` x ) = y -> y e. x ) )
5	4	ralimi 2493	. . . . 5 |- ( A. x e. A ( F ` x ) e. x -> A. x e. A ( ( F ` x ) = y -> y e. x ) )
6		rexim 2524	. . . . 5 |- ( A. x e. A ( ( F ` x ) = y -> y e. x ) -> ( E. x e. A ( F ` x ) = y -> E. x e. A y e. x ) )
7	5, 6	syl 14	. . . 4 |- ( A. x e. A ( F ` x ) e. x -> ( E. x e. A ( F ` x ) = y -> E. x e. A y e. x ) )
8	2, 7	sylan9 406	. . 3 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ( y e. ran F -> E. x e. A y e. x ) )
9		eluni2 3735	. . 3 |- ( y e. U. A <-> E. x e. A y e. x )
10	8, 9	syl6ibr 161	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ( y e. ran F -> y e. U. A ) )
11	10	ssrdv 3098	1 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2414   E.wrex 2415   C_ wss 3066   U.cuni 3731   ran crn 4535   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by: (None)