Theorem aceq4 3557

Description: Equivalence of two versions of the Axiom of Choice. The left-hand side is similar to the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC.

Assertion

Ref	Expression
aceq4	|- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))

Distinct variable group(s):   x,f,z

Proof of Theorem aceq4

Step	Hyp	Ref	Expression
1		aceq3 3556	. 2 |- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.fA.z e. x (-. z = (/) -> (f` z) e. z))
2		fveq1 2831	. . . . . . . . 9 |- (f = y -> (f` z) = (y` z))
3	2	eleq1d 1155	. . . . . . . 8 |- (f = y -> ((f` z) e. z <-> (y` z) e. z))
4	3	imbi2d 464	. . . . . . 7 |- (f = y -> ((-. z = (/) -> (f` z) e. z) <-> (-. z = (/) -> (y` z) e. z)))
5	4	biraldv 1219	. . . . . 6 |- (f = y -> (A.z e. x (-. z = (/) -> (f` z) e. z) <-> A.z e. x (-. z = (/) -> (y` z) e. z)))
6	5	cbvexv 973	. . . . 5 |- (E.fA.z e. x (-. z = (/) -> (f` z) e. z) <-> E.yA.z e. x (-. z = (/) -> (y` z) e. z))
7		fveq2 2832	. . . . . . . . . . . . 13 |- (w = z -> (y` w) = (y` z))
8		cleqid 1102	. . . . . . . . . . . . 13 |- {<.w, v>. | (w e. x /\ v = (y` w))} = {<.w, v>. | (w e. x /\ v = (y` w))}
9		fvex 2838	. . . . . . . . . . . . 13 |- (y` z) e. V
10	7, 8, 9	fvopab4 2871	. . . . . . . . . . . 12 |- (z e. x -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) = (y` z))
11	10	eleq1d 1155	. . . . . . . . . . 11 |- (z e. x -> (({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z <-> (y` z) e. z))
12	11	imbi2d 464	. . . . . . . . . 10 |- (z e. x -> ((-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z) <-> (-. z = (/) -> (y` z) e. z)))
13	12	birala 1228	. . . . . . . . 9 |- (A.z e. x (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z) <-> A.z e. x (-. z = (/) -> (y` z) e. z))
14	13	anbi2i 367	. . . . . . . 8 |- (({<.w, v>. | (w e. x /\ v = (y` w))} Fn x /\ A.z e. x (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z)) <-> ({<.w, v>. | (w e. x /\ v = (y` w))} Fn x /\ A.z e. x (-. z = (/) -> (y` z) e. z)))
15		fvex 2838	. . . . . . . . 9 |- (y` w) e. V
16	15, 8	fnopab2 2747	. . . . . . . 8 |- {<.w, v>. | (w e. x /\ v = (y` w))} Fn x
17	14, 16	mpbiran 547	. . . . . . 7 |- (({<.w, v>. | (w e. x /\ v = (y` w))} Fn x /\ A.z e. x (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z)) <-> A.z e. x (-. z = (/) -> (y` z) e. z))
18		visset 1350	. . . . . . . . 9 |- x e. V
19		moeq 1431	. . . . . . . . . 10 |- E*v v = (y` w)
20	19	a1i 7	. . . . . . . . 9 |- (w e. x -> E*v v = (y` w))
21	18, 20	funopabex 2742	. . . . . . . 8 |- {<.w, v>. | (w e. x /\ v = (y` w))} e. V
22		fneq1 2718	. . . . . . . . 9 |- (f = {<.w, v>. | (w e. x /\ v = (y` w))} -> (f Fn x <-> {<.w, v>. | (w e. x /\ v = (y` w))} Fn x))
23		fveq1 2831	. . . . . . . . . . . 12 |- (f = {<.w, v>. | (w e. x /\ v = (y` w))} -> (f` z) = ({<.w, v>. | (w e. x /\ v = (y` w))}` z))
24	23	eleq1d 1155	. . . . . . . . . . 11 |- (f = {<.w, v>. | (w e. x /\ v = (y` w))} -> ((f` z) e. z <-> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z))
25	24	imbi2d 464	. . . . . . . . . 10 |- (f = {<.w, v>. | (w e. x /\ v = (y` w))} -> ((-. z = (/) -> (f` z) e. z) <-> (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z)))
26	25	biraldv 1219	. . . . . . . . 9 |- (f = {<.w, v>. | (w e. x /\ v = (y` w))} -> (A.z e. x (-. z = (/) -> (f` z) e. z) <-> A.z e. x (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z)))
27	22, 26	anbi12d 476	. . . . . . . 8 |- (f = {<.w, v>. | (w e. x /\ v = (y` w))} -> ((f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)) <-> ({<.w, v>. | (w e. x /\ v = (y` w))} Fn x /\ A.z e. x (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z))))
28	21, 27	cla4ev 1401	. . . . . . 7 |- (({<.w, v>. | (w e. x /\ v = (y` w))} Fn x /\ A.z e. x (-. z = (/) -> ({<.w, v>. | (w e. x /\ v = (y` w))}` z) e. z)) -> E.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))
29	17, 28	sylbir 176	. . . . . 6 |- (A.z e. x (-. z = (/) -> (y` z) e. z) -> E.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))
30	29	19.23aiv 952	. . . . 5 |- (E.yA.z e. x (-. z = (/) -> (y` z) e. z) -> E.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))
31	6, 30	sylbi 174	. . . 4 |- (E.fA.z e. x (-. z = (/) -> (f` z) e. z) -> E.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))
32		pm3.27 260	. . . . 5 |- ((f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)) -> A.z e. x (-. z = (/) -> (f` z) e. z))
33	32	19.22i 723	. . . 4 |- (E.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)) -> E.fA.z e. x (-. z = (/) -> (f` z) e. z))
34	31, 33	impbi 139	. . 3 |- (E.fA.z e. x (-. z = (/) -> (f` z) e. z) <-> E.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))
35	34	bial 695	. 2 |- (A.xE.fA.z e. x (-. z = (/) -> (f` z) e. z) <-> A.xE.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))
36	1, 35	bitr 151	1 |- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.f(f Fn x /\ A.z e. x (-. z = (/) -> (f` z) e. z)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  E*wmo 1008   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487  (/)c0 1707  {copab 2055  dom cdm 2410   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  aceq5 3563  ac5 3573

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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