Theorem ac6 5917

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 larger 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 5918, 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 1884	. . 3 |- {y e. B | ph} = {y e. B | ph}
4		eqeq1 1890	. . . . 5 |- (w = z -> (w = {y e. B | ph} <-> z = {y e. B | ph}))
5	4	anbi2d 678	. . . 4 |- (w = z -> ((x e. A /\ w = {y e. B | ph}) <-> (x e. A /\ z = {y e. B | ph})))
6	5	cbvopab2v 3408	. . 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 5916	. 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 2414	. . . . . 6 |- ((f` x) e. {y e. B | ph} <-> ((f` x) e. B /\ ps))
10	9	simprbi 353	. . . . 5 |- ((f` x) e. {y e. B | ph} -> ps)
11	10	ralimi 2168	. . . 4 |- (A.x e. A (f` x) e. {y e. B | ph} -> A.x e. A ps)
12	11	anim2i 362	. . 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	eximi 1387	. 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 12	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292  {copab 3395  -->wf 3994  ` cfv 3998

This theorem is referenced by:  ac6s 5918  projlem17 10835  osumlem5 11217  surjsec2 14467

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014

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