Theorem chfnrn 5625

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 5561	. . . . 5 |- ( F Fn A -> ( y e. ran F <-> E. x e. A ( F ` x ) = y ) )
2	1	biimpd 144	. . . 4 |- ( F Fn A -> ( y e. ran F -> E. x e. A ( F ` x ) = y ) )
3		eleq1 2240	. . . . . . 7 |- ( ( F ` x ) = y -> ( ( F ` x ) e. x <-> y e. x ) )
4	3	biimpcd 159	. . . . . 6 |- ( ( F ` x ) e. x -> ( ( F ` x ) = y -> y e. x ) )
5	4	ralimi 2540	. . . . 5 |- ( A. x e. A ( F ` x ) e. x -> A. x e. A ( ( F ` x ) = y -> y e. x ) )
6		rexim 2571	. . . . 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 409	. . 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 3813	. . 3 |- ( y e. U. A <-> E. x e. A y e. x )
10	8, 9	syl6ibr 162	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ( y e. ran F -> y e. U. A ) )
11	10	ssrdv 3161	1 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   C_ wss 3129   U.cuni 3809   ran crn 4626   Fn wfn 5209   ` cfv 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5176  df-fun 5216  df-fn 5217  df-fv 5222

This theorem is referenced by: (None)