Theorem ac5 4724

Description: An Axiom of Choice equivalent: there exists a function f (called a choice function) with domain A that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that f be a function is not necessary; see ac4 4722.

Hypothesis

Ref	Expression
ac5.1	|- A e. V

Assertion

Ref	Expression
ac5	|- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))

Distinct variable group:   x,f,A

Proof of Theorem ac5

Step	Hyp	Ref	Expression
1		ac5.1	. 2 |- A e. V
2		fneq2 3569	. . . 4 |- (y = A -> (f Fn y <-> f Fn A))
3		raleq1 1778	. . . 4 |- (y = A -> (A.x e. y (x =/= (/) -> (f` x) e. x) <-> A.x e. A (x =/= (/) -> (f` x) e. x)))
4	2, 3	anbi12d 626	. . 3 |- (y = A -> ((f Fn y /\ A.x e. y (x =/= (/) -> (f` x) e. x)) <-> (f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))))
5	4	exbidv 1274	. 2 |- (y = A -> (E.f(f Fn y /\ A.x e. y (x =/= (/) -> (f` x) e. x)) <-> E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))))
6		aceq3 4705	. . . . 5 |- (A.yE.f(f (_ y /\ f Fn dom y) <-> A.yE.fA.x e. y (x =/= (/) -> (f` x) e. x))
7		ac4 4722	. . . . 5 |- E.fA.x e. y (x =/= (/) -> (f` x) e. x)
8	6, 7	mpgbir 985	. . . 4 |- A.yE.f(f (_ y /\ f Fn dom y)
9		aceq4 4706	. . . 4 |- (A.yE.f(f (_ y /\ f Fn dom y) <-> A.yE.f(f Fn y /\ A.x e. y (x =/= (/) -> (f` x) e. x)))
10	8, 9	mpbi 189	. . 3 |- A.yE.f(f Fn y /\ A.x e. y (x =/= (/) -> (f` x) e. x))
11	10	a4i 979	. 2 |- E.f(f Fn y /\ A.x e. y (x =/= (/) -> (f` x) e. x))
12	1, 5, 11	vtocl 1833	1 |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  A.wral 1637  Vcvv 1802   (_ wss 2037  (/)c0 2270  dom cdm 3160   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  ac5b 4725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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