Theorem ac5 4676

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 4674.

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 3523	. . . 4 |- (y = A -> (f Fn y <-> f Fn A))
3		raleq1 1762	. . . 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 1261	. 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 4657	. . . . 5 |- (A.yE.f(f (_ y /\ f Fn dom y) <-> A.yE.fA.x e. y (x =/= (/) -> (f` x) e. x))
7		ac4 4674	. . . . 5 |- E.fA.x e. y (x =/= (/) -> (f` x) e. x)
8	6, 7	mpgbir 964	. . . 4 |- A.yE.f(f (_ y /\ f Fn dom y)
9		aceq4 4658	. . . 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 958	. 2 |- E.f(f Fn y /\ A.x e. y (x =/= (/) -> (f` x) e. x))
12	1, 5, 11	vtocl 1817	1 |- E.f(f Fn A /\ A.x e. A (x =/= (/) -> (f` x) e. x))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105   =/= wne 1561  A.wral 1621  Vcvv 1786   (_ wss 2018  (/)c0 2251  dom cdm 3133   Fn wfn 3140  ` cfv 3145

This theorem is referenced by:  ac5b 4677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161

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