Theorem chfnrn 5673

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 5608	. . . . 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 2259	. . . . . . 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 2560	. . . . 5 |- ( A. x e. A ( F ` x ) e. x -> A. x e. A ( ( F ` x ) = y -> y e. x ) )
6		rexim 2591	. . . . 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 3843	. . 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 3189	1 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   U.cuni 3839   ran crn 4664   Fn wfn 5253   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by: (None)