Theorem chfnrn 3802

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.

Assertion

Ref	Expression
chfnrn	|- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F (_ U.A)

Distinct variable groups:   x,A   x,F

Proof of Theorem chfnrn

Step	Hyp	Ref	Expression
1		fvelrnb 3760	. . . . 5 |- (F Fn A -> (y e. ran F <-> E.x e. A (F` x) = y))
2	1	biimpd 153	. . . 4 |- (F Fn A -> (y e. ran F -> E.x e. A (F` x) = y))
3		hbra1 1687	. . . . 5 |- (A.x e. A (F` x) e. x -> A.xA.x e. A (F` x) e. x)
4		ra4 1694	. . . . . 6 |- (A.x e. A (F` x) e. x -> (x e. A -> (F` x) e. x))
5		eleq1 1534	. . . . . . 7 |- ((F` x) = y -> ((F` x) e. x <-> y e. x))
6	5	biimpcd 155	. . . . . 6 |- ((F` x) e. x -> ((F` x) = y -> y e. x))
7	4, 6	syl6 22	. . . . 5 |- (A.x e. A (F` x) e. x -> (x e. A -> ((F` x) = y -> y e. x)))
8	3, 7	r19.22d 1735	. . . 4 |- (A.x e. A (F` x) e. x -> (E.x e. A (F` x) = y -> E.x e. A y e. x))
9	2, 8	sylan9 468	. . 3 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> (y e. ran F -> E.x e. A y e. x))
10		eluni2 2507	. . 3 |- (y e. U.A <-> E.x e. A y e. x)
11	9, 10	syl6ibr 213	. 2 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> (y e. ran F -> y e. U.A))
12	11	ssrdv 2070	1 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F (_ U.A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047  U.cuni 2503  ran crn 3171   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  ac5b 4753

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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