Theorem chfnrn 5596

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 5534	. . . . 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 2229	. . . . . . 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 2529	. . . . 5 |- ( A. x e. A ( F ` x ) e. x -> A. x e. A ( ( F ` x ) = y -> y e. x ) )
6		rexim 2560	. . . . 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 407	. . 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 3793	. . 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 3148	1 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. x ) -> ran F C_ U. A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   C_ wss 3116   U.cuni 3789   ran crn 4605   Fn wfn 5183   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by: (None)