Theorem insklem 4304

Description: Lemma for ins2kexg 4305 and ins3kexg 4306. Equality for subsets of (~P₁ 1_c X._k (_V X._k _V)). (Contributed by SF, 14-Jan-2015.)

Hypotheses

Ref	Expression
insklem.1	|- A C_ (~P1 1c X.k (_V X.k _V))
insklem.2	|- B C_ (~P1 1c X.k (_V X.k _V))

Assertion

Ref	Expression
insklem	|- (A = B <-> A.xA.yA.z(<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B))

Distinct variable groups:   x,A,y,z   x,B,y,z

Proof of Theorem insklem

Dummy variables w t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		insklem.1	. . 3 |- A C_ (~P1 1c X.k (_V X.k _V))
2		insklem.2	. . 3 |- B C_ (~P1 1c X.k (_V X.k _V))
3		ssofeq 4077	. . 3 |- ((A C_ (~P1 1c X.k (_V X.k _V)) /\ B C_ (~P1 1c X.k (_V X.k _V))) -> (A = B <-> A.w e. (~P1 1c X.k (_V X.k _V))(w e. A <-> w e. B)))
4	1, 2, 3	mp2an 653	. 2 |- (A = B <-> A.w e. (~P1 1c X.k (_V X.k _V))(w e. A <-> w e. B))
5		19.23v 1891	. . . . 5 |- (A.x(E.yE.z w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> (E.xE.yE.z w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
6		19.23vv 1892	. . . . . 6 |- (A.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> (E.yE.z w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
7	6	albii 1566	. . . . 5 |- (A.xA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> A.x(E.yE.z w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
8		19.42vv 1907	. . . . . . . . . 10 |- (E.yE.z(t e. ~P1 1c /\ u = <<y, z>>) <-> (t e. ~P1 1c /\ E.yE.z u = <<y, z>>))
9	8	anbi2i 675	. . . . . . . . 9 |- ((w = <<t, u>> /\ E.yE.z(t e. ~P1 1c /\ u = <<y, z>>)) <-> (w = <<t, u>> /\ (t e. ~P1 1c /\ E.yE.z u = <<y, z>>)))
10		19.42vv 1907	. . . . . . . . 9 |- (E.yE.z(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)) <-> (w = <<t, u>> /\ E.yE.z(t e. ~P1 1c /\ u = <<y, z>>)))
11		elvvk 4207	. . . . . . . . . . 11 |- (u e. (_V X.k _V) <-> E.yE.z u = <<y, z>>)
12	11	anbi2i 675	. . . . . . . . . 10 |- ((t e. ~P1 1c /\ u e. (_V X.k _V)) <-> (t e. ~P1 1c /\ E.yE.z u = <<y, z>>))
13	12	anbi2i 675	. . . . . . . . 9 |- ((w = <<t, u>> /\ (t e. ~P1 1c /\ u e. (_V X.k _V))) <-> (w = <<t, u>> /\ (t e. ~P1 1c /\ E.yE.z u = <<y, z>>)))
14	9, 10, 13	3bitr4ri 269	. . . . . . . 8 |- ((w = <<t, u>> /\ (t e. ~P1 1c /\ u e. (_V X.k _V))) <-> E.yE.z(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
15	14	2exbii 1583	. . . . . . 7 |- (E.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u e. (_V X.k _V))) <-> E.tE.uE.yE.z(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
16		elxpk 4196	. . . . . . 7 |- (w e. (~P1 1c X.k (_V X.k _V)) <-> E.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u e. (_V X.k _V))))
17		exrot3 1744	. . . . . . . 8 |- (E.xE.yE.z w = <<{{x}}, <<y, z>>>> <-> E.yE.zE.x w = <<{{x}}, <<y, z>>>>)
18		exancom 1586	. . . . . . . . . . . 12 |- (E.t(w = <<t, <<y, z>>>> /\ t e. ~P1 1c) <-> E.t(t e. ~P1 1c /\ w = <<t, <<y, z>>>>))
19		elpw11c 4147	. . . . . . . . . . . . . . 15 |- (t e. ~P1 1c <-> E.x t = {{x}})
20	19	anbi1i 676	. . . . . . . . . . . . . 14 |- ((t e. ~P1 1c /\ w = <<t, <<y, z>>>>) <-> (E.x t = {{x}} /\ w = <<t, <<y, z>>>>))
21		19.41v 1901	. . . . . . . . . . . . . 14 |- (E.x(t = {{x}} /\ w = <<t, <<y, z>>>>) <-> (E.x t = {{x}} /\ w = <<t, <<y, z>>>>))
22	20, 21	bitr4i 243	. . . . . . . . . . . . 13 |- ((t e. ~P1 1c /\ w = <<t, <<y, z>>>>) <-> E.x(t = {{x}} /\ w = <<t, <<y, z>>>>))
23	22	exbii 1582	. . . . . . . . . . . 12 |- (E.t(t e. ~P1 1c /\ w = <<t, <<y, z>>>>) <-> E.tE.x(t = {{x}} /\ w = <<t, <<y, z>>>>))
24	18, 23	bitri 240	. . . . . . . . . . 11 |- (E.t(w = <<t, <<y, z>>>> /\ t e. ~P1 1c) <-> E.tE.x(t = {{x}} /\ w = <<t, <<y, z>>>>))
25		ancom 437	. . . . . . . . . . . . . . 15 |- ((t e. ~P1 1c /\ u = <<y, z>>) <-> (u = <<y, z>> /\ t e. ~P1 1c))
26	25	anbi2i 675	. . . . . . . . . . . . . 14 |- ((w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)) <-> (w = <<t, u>> /\ (u = <<y, z>> /\ t e. ~P1 1c)))
27		an12 772	. . . . . . . . . . . . . 14 |- ((w = <<t, u>> /\ (u = <<y, z>> /\ t e. ~P1 1c)) <-> (u = <<y, z>> /\ (w = <<t, u>> /\ t e. ~P1 1c)))
28	26, 27	bitri 240	. . . . . . . . . . . . 13 |- ((w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)) <-> (u = <<y, z>> /\ (w = <<t, u>> /\ t e. ~P1 1c)))
29	28	2exbii 1583	. . . . . . . . . . . 12 |- (E.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)) <-> E.tE.u(u = <<y, z>> /\ (w = <<t, u>> /\ t e. ~P1 1c)))
30		opkex 4113	. . . . . . . . . . . . . 14 |- <<y, z>> e. _V
31		opkeq2 4060	. . . . . . . . . . . . . . . 16 |- (u = <<y, z>> -> <<t, u>> = <<t, <<y, z>>>>)
32	31	eqeq2d 2364	. . . . . . . . . . . . . . 15 |- (u = <<y, z>> -> (w = <<t, u>> <-> w = <<t, <<y, z>>>>))
33	32	anbi1d 685	. . . . . . . . . . . . . 14 |- (u = <<y, z>> -> ((w = <<t, u>> /\ t e. ~P1 1c) <-> (w = <<t, <<y, z>>>> /\ t e. ~P1 1c)))
34	30, 33	ceqsexv 2894	. . . . . . . . . . . . 13 |- (E.u(u = <<y, z>> /\ (w = <<t, u>> /\ t e. ~P1 1c)) <-> (w = <<t, <<y, z>>>> /\ t e. ~P1 1c))
35	34	exbii 1582	. . . . . . . . . . . 12 |- (E.tE.u(u = <<y, z>> /\ (w = <<t, u>> /\ t e. ~P1 1c)) <-> E.t(w = <<t, <<y, z>>>> /\ t e. ~P1 1c))
36	29, 35	bitri 240	. . . . . . . . . . 11 |- (E.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)) <-> E.t(w = <<t, <<y, z>>>> /\ t e. ~P1 1c))
37		snex 4111	. . . . . . . . . . . . . 14 |- {{x}} e. _V
38		opkeq1 4059	. . . . . . . . . . . . . . 15 |- (t = {{x}} -> <<t, <<y, z>>>> = <<{{x}}, <<y, z>>>>)
39	38	eqeq2d 2364	. . . . . . . . . . . . . 14 |- (t = {{x}} -> (w = <<t, <<y, z>>>> <-> w = <<{{x}}, <<y, z>>>>))
40	37, 39	ceqsexv 2894	. . . . . . . . . . . . 13 |- (E.t(t = {{x}} /\ w = <<t, <<y, z>>>>) <-> w = <<{{x}}, <<y, z>>>>)
41	40	exbii 1582	. . . . . . . . . . . 12 |- (E.xE.t(t = {{x}} /\ w = <<t, <<y, z>>>>) <-> E.x w = <<{{x}}, <<y, z>>>>)
42		excom 1741	. . . . . . . . . . . 12 |- (E.xE.t(t = {{x}} /\ w = <<t, <<y, z>>>>) <-> E.tE.x(t = {{x}} /\ w = <<t, <<y, z>>>>))
43	41, 42	bitr3i 242	. . . . . . . . . . 11 |- (E.x w = <<{{x}}, <<y, z>>>> <-> E.tE.x(t = {{x}} /\ w = <<t, <<y, z>>>>))
44	24, 36, 43	3bitr4ri 269	. . . . . . . . . 10 |- (E.x w = <<{{x}}, <<y, z>>>> <-> E.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
45	44	2exbii 1583	. . . . . . . . 9 |- (E.yE.zE.x w = <<{{x}}, <<y, z>>>> <-> E.yE.zE.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
46		exrot4 1745	. . . . . . . . 9 |- (E.yE.zE.tE.u(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)) <-> E.tE.uE.yE.z(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
47	45, 46	bitri 240	. . . . . . . 8 |- (E.yE.zE.x w = <<{{x}}, <<y, z>>>> <-> E.tE.uE.yE.z(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
48	17, 47	bitri 240	. . . . . . 7 |- (E.xE.yE.z w = <<{{x}}, <<y, z>>>> <-> E.tE.uE.yE.z(w = <<t, u>> /\ (t e. ~P1 1c /\ u = <<y, z>>)))
49	15, 16, 48	3bitr4i 268	. . . . . 6 |- (w e. (~P1 1c X.k (_V X.k _V)) <-> E.xE.yE.z w = <<{{x}}, <<y, z>>>>)
50	49	imbi1i 315	. . . . 5 |- ((w e. (~P1 1c X.k (_V X.k _V)) -> (w e. A <-> w e. B)) <-> (E.xE.yE.z w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
51	5, 7, 50	3bitr4ri 269	. . . 4 |- ((w e. (~P1 1c X.k (_V X.k _V)) -> (w e. A <-> w e. B)) <-> A.xA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
52	51	albii 1566	. . 3 |- (A.w(w e. (~P1 1c X.k (_V X.k _V)) -> (w e. A <-> w e. B)) <-> A.wA.xA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
53		df-ral 2619	. . 3 |- (A.w e. (~P1 1c X.k (_V X.k _V))(w e. A <-> w e. B) <-> A.w(w e. (~P1 1c X.k (_V X.k _V)) -> (w e. A <-> w e. B)))
54		alcom 1737	. . . 4 |- (A.wA.xA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> A.xA.wA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
55		alrot3 1738	. . . . 5 |- (A.wA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> A.yA.zA.w(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
56	55	albii 1566	. . . 4 |- (A.xA.wA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> A.xA.yA.zA.w(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
57		opkex 4113	. . . . . . 7 |- <<{{x}}, <<y, z>>>> e. _V
58		eleq1 2413	. . . . . . . 8 |- (w = <<{{x}}, <<y, z>>>> -> (w e. A <-> <<{{x}}, <<y, z>>>> e. A))
59		eleq1 2413	. . . . . . . 8 |- (w = <<{{x}}, <<y, z>>>> -> (w e. B <-> <<{{x}}, <<y, z>>>> e. B))
60	58, 59	bibi12d 312	. . . . . . 7 |- (w = <<{{x}}, <<y, z>>>> -> ((w e. A <-> w e. B) <-> (<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B)))
61	57, 60	ceqsalv 2885	. . . . . 6 |- (A.w(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> (<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B))
62	61	albii 1566	. . . . 5 |- (A.zA.w(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> A.z(<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B))
63	62	2albii 1567	. . . 4 |- (A.xA.yA.zA.w(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)) <-> A.xA.yA.z(<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B))
64	54, 56, 63	3bitrri 263	. . 3 |- (A.xA.yA.z(<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B) <-> A.wA.xA.yA.z(w = <<{{x}}, <<y, z>>>> -> (w e. A <-> w e. B)))
65	52, 53, 64	3bitr4i 268	. 2 |- (A.w e. (~P1 1c X.k (_V X.k _V))(w e. A <-> w e. B) <-> A.xA.yA.z(<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B))
66	4, 65	bitri 240	1 |- (A = B <-> A.xA.yA.z(<<{{x}}, <<y, z>>>> e. A <-> <<{{x}}, <<y, z>>>> e. B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  A.wral 2614  _Vcvv 2859   C_ wss 3257  {csn 3737  <<copk 4057  1_cc1c 4134  ~P₁ cpw1 4135   X._k cxpk 4174

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185

This theorem is referenced by:  ins2kexg  4305  ins3kexg  4306