Theorem chfnrn 5820

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 5750	. . . . 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 2301	. . . . . . 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 2613	. . . . 5 |- ( A. x e. A ( F ` x ) e. x -> A. x e. A ( ( F ` x ) = y -> y e. x ) )
6		rexim 2644	. . . . 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 413	. . 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 3939	. . 3 |- ( y e. U. A <-> E. x e. A y e. x )
10	8, 9	imbitrrdi 162	. 2 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ( y e. ran F -> y e. U. A ) )
11	10	ssrdv 3254	1 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   C_ wss 3220   U.cuni 3935   ran crn 4775   Fn wfn 5372   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by: (None)