Theorem ac6 4765

Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a large set B, where ph depends on x (the natural number) and y (to specify a member of B). A stronger version of this theorem, ac6s 4766, allows B to be a proper class.

Hypotheses

Ref	Expression
ac6.1	|- A e. V
ac6.2	|- B e. V
ac6.3	|- (y = (f` x) -> (ph <-> ps))

Assertion

Ref	Expression
ac6	|- (A.x e. A E.y e. B ph -> E.f(f:A-->B /\ A.x e. A ps))

Distinct variable groups:   x,f,y,A   B,f,x,y   ph,f   ps,y

Proof of Theorem ac6

Step	Hyp	Ref	Expression
1		ac6.1	. . 3 |- A e. V
2		ac6.2	. . 3 |- B e. V
3		eqid 1478	. . 3 |- {y e. B | ph} = {y e. B | ph}
4		eqeq1 1484	. . . . 5 |- (w = z -> (w = {y e. B | ph} <-> z = {y e. B | ph}))
5	4	anbi2d 618	. . . 4 |- (w = z -> ((x e. A /\ w = {y e. B | ph}) <-> (x e. A /\ z = {y e. B | ph})))
6	5	cbvopab2v 2682	. . 3 |- {<.x, w>. | (x e. A /\ w = {y e. B | ph})} = {<.x, z>. | (x e. A /\ z = {y e. B | ph})}
7	1, 2, 3, 6	ac6lem 4764	. 2 |- (A.x e. A E.y e. B ph -> E.f(f:A-->B /\ A.x e. A (f` x) e. {y e. B | ph}))
8		ac6.3	. . . . . . 7 |- (y = (f` x) -> (ph <-> ps))
9	8	elrab 1908	. . . . . 6 |- ((f` x) e. {y e. B | ph} <-> ((f` x) e. B /\ ps))
10	9	pm3.27bi 326	. . . . 5 |- ((f` x) e. {y e. B | ph} -> ps)
11	10	r19.20si 1709	. . . 4 |- (A.x e. A (f` x) e. {y e. B | ph} -> A.x e. A ps)
12	11	anim2i 335	. . 3 |- ((f:A-->B /\ A.x e. A (f` x) e. {y e. B | ph}) -> (f:A-->B /\ A.x e. A ps))
13	12	19.22i 1042	. 2 |- (E.f(f:A-->B /\ A.x e. A (f` x) e. {y e. B | ph}) -> E.f(f:A-->B /\ A.x e. A ps))
14	7, 13	syl 10	1 |- (A.x e. A E.y e. B ph -> E.f(f:A-->B /\ A.x e. A ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  A.wral 1648  E.wrex 1649  {crab 1651  Vcvv 1814  {copab 2671  -->wf 3184  ` cfv 3188

This theorem is referenced by:  ac6s 4766  projlem17 9197  osumlem5 9577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204

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