Theorem chfnrn 3793

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 3751	. . . . 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 1684	. . . . 5 |- (A.x e. A (F` x) e. x -> A.xA.x e. A (F` x) e. x)
4		ra4 1691	. . . . . 6 |- (A.x e. A (F` x) e. x -> (x e. A -> (F` x) e. x))
5		eleq1 1531	. . . . . . 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 1732	. . . 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 2502	. . 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 2066	1 |- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F (_ U.A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643   (_ wss 2043  U.cuni 2498  ran crn 3166   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  ac5b 4733

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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