Theorem euexeqOLD 3826

Description: Two ways to express single-valuedness of a class expression A(x).

Hypothesis

Ref	Expression
euexeqOLD.1	|- A e. _V

Assertion

Ref	Expression
euexeqOLD	|- (E!yE.x y = A <-> E!yA.x y = A)

Distinct variable groups:   x,y   y,A

Proof of Theorem euexeqOLD

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . . 8 |- z e. _V
2	1	biantrur 794	. . . . . . 7 |- (E.x z = A <-> (z e. _V /\ E.x z = A))
3	2	eubii 1780	. . . . . 6 |- (E!zE.x z = A <-> E!z(z e. _V /\ E.x z = A))
4		ax-17 1317	. . . . . . 7 |- (E.x y = A -> A.zE.x y = A)
5		ax-17 1317	. . . . . . 7 |- (E.x z = A -> A.yE.x z = A)
6		eqeq1 1890	. . . . . . . 8 |- (y = z -> (y = A <-> z = A))
7	6	exbidv 1657	. . . . . . 7 |- (y = z -> (E.x y = A <-> E.x z = A))
8	4, 5, 7	cbveu 1785	. . . . . 6 |- (E!yE.x y = A <-> E!zE.x z = A)
9		df-reu 2111	. . . . . 6 |- (E!z e. _V E.x z = A <-> E!z(z e. _V /\ E.x z = A))
10	3, 8, 9	3bitr4i 200	. . . . 5 |- (E!yE.x y = A <-> E!z e. _V E.x z = A)
11		reucl 3213	. . . . 5 |- (E!z e. _V E.x z = A -> U.{z e. _V | E.x z = A} e. _V)
12	10, 11	sylbi 216	. . . 4 |- (E!yE.x y = A -> U.{z e. _V | E.x z = A} e. _V)
13		hbe1 1363	. . . . . 6 |- (E.x y = A -> A.xE.x y = A)
14	13	hbeu 1782	. . . . 5 |- (E!yE.x y = A -> A.xE!yE.x y = A)
15		eqid 1884	. . . . . 6 |- A = A
16		eqeq1 1890	. . . . . . . . . . . . 13 |- (y = w -> (y = A <-> w = A))
17	16	exbidv 1657	. . . . . . . . . . . 12 |- (y = w -> (E.x y = A <-> E.x w = A))
18	17	eu4 1806	. . . . . . . . . . 11 |- (E!yE.x y = A <-> (E.yE.x y = A /\ A.yA.w((E.x y = A /\ E.x w = A) -> y = w)))
19		ax-17 1317	. . . . . . . . . . . . . . . . . . 19 |- (v e. y -> A.x v e. y)
20		hbe1 1363	. . . . . . . . . . . . . . . . . . . . 21 |- (E.x z = A -> A.xE.x z = A)
21		ax-17 1317	. . . . . . . . . . . . . . . . . . . . 21 |- (v e. _V -> A.x v e. _V)
22	20, 21	hbrab 2258	. . . . . . . . . . . . . . . . . . . 20 |- (v e. {z e. _V | E.x z = A} -> A.x v e. {z e. _V | E.x z = A})
23	22	hbuni 3183	. . . . . . . . . . . . . . . . . . 19 |- (v e. U.{z e. _V | E.x z = A} -> A.x v e. U.{z e. _V | E.x z = A})
24	19, 23	hbeq 1995	. . . . . . . . . . . . . . . . . 18 |- (y = U.{z e. _V | E.x z = A} -> A.x y = U.{z e. _V | E.x z = A})
25		eqeq1 1890	. . . . . . . . . . . . . . . . . 18 |- (y = U.{z e. _V | E.x z = A} -> (y = A <-> U.{z e. _V | E.x z = A} = A))
26	24, 25	exbid 1460	. . . . . . . . . . . . . . . . 17 |- (y = U.{z e. _V | E.x z = A} -> (E.x y = A <-> E.xU.{z e. _V | E.x z = A} = A))
27	26	anbi1d 679	. . . . . . . . . . . . . . . 16 |- (y = U.{z e. _V | E.x z = A} -> ((E.x y = A /\ E.x w = A) <-> (E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A)))
28		eqeq1 1890	. . . . . . . . . . . . . . . 16 |- (y = U.{z e. _V | E.x z = A} -> (y = w <-> U.{z e. _V | E.x z = A} = w))
29	27, 28	imbi12d 688	. . . . . . . . . . . . . . 15 |- (y = U.{z e. _V | E.x z = A} -> (((E.x y = A /\ E.x w = A) -> y = w) <-> ((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
30	29	albidv 1656	. . . . . . . . . . . . . 14 |- (y = U.{z e. _V | E.x z = A} -> (A.w((E.x y = A /\ E.x w = A) -> y = w) <-> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
31	30	cla4gv 2364	. . . . . . . . . . . . 13 |- (U.{z e. _V | E.x z = A} e. _V -> (A.yA.w((E.x y = A /\ E.x w = A) -> y = w) -> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
32	31	com12 14	. . . . . . . . . . . 12 |- (A.yA.w((E.x y = A /\ E.x w = A) -> y = w) -> (U.{z e. _V | E.x z = A} e. _V -> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
33	32	adantl 424	. . . . . . . . . . 11 |- ((E.yE.x y = A /\ A.yA.w((E.x y = A /\ E.x w = A) -> y = w)) -> (U.{z e. _V | E.x z = A} e. _V -> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
34	18, 33	sylbi 216	. . . . . . . . . 10 |- (E!yE.x y = A -> (U.{z e. _V | E.x z = A} e. _V -> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
35	12, 34	mpd 29	. . . . . . . . 9 |- (E!yE.x y = A -> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w))
36		visset 2295	. . . . . . . . . . . . . . . 16 |- y e. _V
37	36	biantrur 794	. . . . . . . . . . . . . . 15 |- (E.x y = A <-> (y e. _V /\ E.x y = A))
38	37	eubii 1780	. . . . . . . . . . . . . 14 |- (E!yE.x y = A <-> E!y(y e. _V /\ E.x y = A))
39		df-reu 2111	. . . . . . . . . . . . . 14 |- (E!y e. _V E.x y = A <-> E!y(y e. _V /\ E.x y = A))
40	38, 39	bitr4i 193	. . . . . . . . . . . . 13 |- (E!yE.x y = A <-> E!y e. _V E.x y = A)
41	7, 26	reuuni3 3812	. . . . . . . . . . . . 13 |- (E!y e. _V E.x y = A -> E.xU.{z e. _V | E.x z = A} = A)
42	40, 41	sylbi 216	. . . . . . . . . . . 12 |- (E!yE.x y = A -> E.xU.{z e. _V | E.x z = A} = A)
43	42	biantrurd 796	. . . . . . . . . . 11 |- (E!yE.x y = A -> (E.x w = A <-> (E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A)))
44	43	imbi1d 675	. . . . . . . . . 10 |- (E!yE.x y = A -> ((E.x w = A -> U.{z e. _V | E.x z = A} = w) <-> ((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
45	44	albidv 1656	. . . . . . . . 9 |- (E!yE.x y = A -> (A.w(E.x w = A -> U.{z e. _V | E.x z = A} = w) <-> A.w((E.xU.{z e. _V | E.x z = A} = A /\ E.x w = A) -> U.{z e. _V | E.x z = A} = w)))
46	35, 45	mpbird 213	. . . . . . . 8 |- (E!yE.x y = A -> A.w(E.x w = A -> U.{z e. _V | E.x z = A} = w))
47		ax-17 1317	. . . . . . . . . . . 12 |- (v e. w -> A.x v e. w)
48	23, 47	hbeq 1995	. . . . . . . . . . 11 |- (U.{z e. _V | E.x z = A} = w -> A.xU.{z e. _V | E.x z = A} = w)
49	48	19.23 1411	. . . . . . . . . 10 |- (A.x(w = A -> U.{z e. _V | E.x z = A} = w) <-> (E.x w = A -> U.{z e. _V | E.x z = A} = w))
50	49	albii 1346	. . . . . . . . 9 |- (A.wA.x(w = A -> U.{z e. _V | E.x z = A} = w) <-> A.w(E.x w = A -> U.{z e. _V | E.x z = A} = w))
51		alcom 1379	. . . . . . . . 9 |- (A.wA.x(w = A -> U.{z e. _V | E.x z = A} = w) <-> A.xA.w(w = A -> U.{z e. _V | E.x z = A} = w))
52	50, 51	bitr3i 192	. . . . . . . 8 |- (A.w(E.x w = A -> U.{z e. _V | E.x z = A} = w) <-> A.xA.w(w = A -> U.{z e. _V | E.x z = A} = w))
53	46, 52	sylib 215	. . . . . . 7 |- (E!yE.x y = A -> A.xA.w(w = A -> U.{z e. _V | E.x z = A} = w))
54		ax-4 1319	. . . . . . 7 |- (A.xA.w(w = A -> U.{z e. _V | E.x z = A} = w) -> A.w(w = A -> U.{z e. _V | E.x z = A} = w))
55		euexeqOLD.1	. . . . . . . 8 |- A e. _V
56		eqeq1 1890	. . . . . . . . 9 |- (w = A -> (w = A <-> A = A))
57		eqeq2 1893	. . . . . . . . 9 |- (w = A -> (U.{z e. _V | E.x z = A} = w <-> U.{z e. _V | E.x z = A} = A))
58	56, 57	imbi12d 688	. . . . . . . 8 |- (w = A -> ((w = A -> U.{z e. _V | E.x z = A} = w) <-> (A = A -> U.{z e. _V | E.x z = A} = A)))
59	55, 58	cla4v 2370	. . . . . . 7 |- (A.w(w = A -> U.{z e. _V | E.x z = A} = w) -> (A = A -> U.{z e. _V | E.x z = A} = A))
60	53, 54, 59	3syl 24	. . . . . 6 |- (E!yE.x y = A -> (A = A -> U.{z e. _V | E.x z = A} = A))
61	15, 60	mpi 55	. . . . 5 |- (E!yE.x y = A -> U.{z e. _V | E.x z = A} = A)
62	14, 61	19.21ai 1345	. . . 4 |- (E!yE.x y = A -> A.xU.{z e. _V | E.x z = A} = A)
63	24, 25	albid 1459	. . . . 5 |- (y = U.{z e. _V | E.x z = A} -> (A.x y = A <-> A.xU.{z e. _V | E.x z = A} = A))
64	63	cla4egv 2365	. . . 4 |- (U.{z e. _V | E.x z = A} e. _V -> (A.xU.{z e. _V | E.x z = A} = A -> E.yA.x y = A))
65	12, 62, 64	sylc 83	. . 3 |- (E!yE.x y = A -> E.yA.x y = A)
66	6	albidv 1656	. . . . . 6 |- (y = z -> (A.x y = A <-> A.x z = A))
67	66	eu4 1806	. . . . 5 |- (E!yA.x y = A <-> (E.yA.x y = A /\ A.yA.z((A.x y = A /\ A.x z = A) -> y = z)))
68		eqtr3 1907	. . . . . . 7 |- ((y = A /\ z = A) -> y = z)
69		ax-4 1319	. . . . . . 7 |- (A.x y = A -> y = A)
70		ax-4 1319	. . . . . . 7 |- (A.x z = A -> z = A)
71	68, 69, 70	syl2an 503	. . . . . 6 |- ((A.x y = A /\ A.x z = A) -> y = z)
72	71	gen2 1329	. . . . 5 |- A.yA.z((A.x y = A /\ A.x z = A) -> y = z)
73	67, 72	mpbiran2 799	. . . 4 |- (E!yA.x y = A <-> E.yA.x y = A)
74	73	bicomi 189	. . 3 |- (E.yA.x y = A <-> E!yA.x y = A)
75	65, 74	sylib 215	. 2 |- (E!yE.x y = A -> E!yA.x y = A)
76		hbeu1 1781	. . . 4 |- (E!yA.x y = A -> A.yE!yA.x y = A)
77		hba1 1350	. . . . . . 7 |- (A.x y = A -> A.xA.x y = A)
78	77	hbeu 1782	. . . . . 6 |- (E!yA.x y = A -> A.xE!yA.x y = A)
79	36	biantrur 794	. . . . . . . . 9 |- (A.x y = A <-> (y e. _V /\ A.x y = A))
80	79	eubii 1780	. . . . . . . 8 |- (E!yA.x y = A <-> E!y(y e. _V /\ A.x y = A))
81		df-reu 2111	. . . . . . . 8 |- (E!y e. _V A.x y = A <-> E!y(y e. _V /\ A.x y = A))
82	80, 81	bitr4i 193	. . . . . . 7 |- (E!yA.x y = A <-> E!y e. _V A.x y = A)
83		hba1 1350	. . . . . . . . . . . . . 14 |- (A.x z = A -> A.xA.x z = A)
84	83, 21	hbrab 2258	. . . . . . . . . . . . 13 |- (v e. {z e. _V | A.x z = A} -> A.x v e. {z e. _V | A.x z = A})
85	84	hbuni 3183	. . . . . . . . . . . 12 |- (v e. U.{z e. _V | A.x z = A} -> A.x v e. U.{z e. _V | A.x z = A})
86	19, 85	hbeq 1995	. . . . . . . . . . 11 |- (y = U.{z e. _V | A.x z = A} -> A.x y = U.{z e. _V | A.x z = A})
87		eqeq1 1890	. . . . . . . . . . 11 |- (y = U.{z e. _V | A.x z = A} -> (y = A <-> U.{z e. _V | A.x z = A} = A))
88	86, 87	albid 1459	. . . . . . . . . 10 |- (y = U.{z e. _V | A.x z = A} -> (A.x y = A <-> A.xU.{z e. _V | A.x z = A} = A))
89	66, 88	reuuni3 3812	. . . . . . . . 9 |- (E!y e. _V A.x y = A -> A.xU.{z e. _V | A.x z = A} = A)
90	89	19.21bi 1408	. . . . . . . 8 |- (E!y e. _V A.x y = A -> U.{z e. _V | A.x z = A} = A)
91	88, 89	syl5cbir 228	. . . . . . . . 9 |- (E!y e. _V A.x y = A -> (y = U.{z e. _V | A.x z = A} -> A.x y = A))
92		eqtr3 1907	. . . . . . . . 9 |- ((y = A /\ U.{z e. _V | A.x z = A} = A) -> y = U.{z e. _V | A.x z = A})
93	91, 92	syl5 20	. . . . . . . 8 |- (E!y e. _V A.x y = A -> ((y = A /\ U.{z e. _V | A.x z = A} = A) -> A.x y = A))
94	90, 93	mpan2d 766	. . . . . . 7 |- (E!y e. _V A.x y = A -> (y = A -> A.x y = A))
95	82, 94	sylbi 216	. . . . . 6 |- (E!yA.x y = A -> (y = A -> A.x y = A))
96	78, 77, 95	19.23ad 1415	. . . . 5 |- (E!yA.x y = A -> (E.x y = A -> A.x y = A))
97		19.2 1377	. . . . 5 |- (A.x y = A -> E.x y = A)
98	96, 97	impbid1 575	. . . 4 |- (E!yA.x y = A -> (E.x y = A <-> A.x y = A))
99	76, 98	eubid 1778	. . 3 |- (E!yA.x y = A -> (E!yE.x y = A <-> E!yA.x y = A))
100	99	ibir 653	. 2 |- (E!yA.x y = A -> E!yE.x y = A)
101	75, 100	impbii 174	1 |- (E!yE.x y = A <-> E!yA.x y = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  E!wreu 2107  {crab 2108  _Vcvv 2292  U.cuni 3177

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-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

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-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-uni 3178

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