Theorem ac6 4755

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 4756, 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 1475	. . 3 |- {y e. B | ph} = {y e. B | ph}
4		eqeq1 1481	. . . . 5 |- (w = z -> (w = {y e. B | ph} <-> z = {y e. B | ph}))
5	4	anbi2d 616	. . . 4 |- (w = z -> ((x e. A /\ w = {y e. B | ph}) <-> (x e. A /\ z = {y e. B | ph})))
6	5	cbvopab2v 2677	. . 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 4754	. 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 1905	. . . . . 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 1706	. . . 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 1040	. 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 956   e. wcel 958  E.wex 980  A.wral 1645  E.wrex 1646  {crab 1648  Vcvv 1811  {copab 2666  -->wf 3178  ` cfv 3182

This theorem is referenced by:  ac6s 4756  projlem17 9202  osumlem5 9582

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-9 965  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-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-reu 1651  df-rab 1652  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-rel 3185  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-f 3194  df-fv 3198

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