Theorem opkelcokg 4261

Description: Membership in a Kuratowski composition. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
opkelcokg	|- ((A e. V /\ B e. W) -> (<<A, B>> e. (C o.k D) <-> E.x(<<A, x>> e. D /\ <<x, B>> e. C)))

Distinct variable groups:   x,A   x,B   x,C   x,D

Allowed substitution hints:   V(x)   W(x)

Proof of Theorem opkelcokg

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2867	. 2 |- (A e. V -> A e. _V)
2		elex 2867	. 2 |- (B e. W -> B e. _V)
3		df-cok 4190	. . . . 5 |- (C o.k D) = (( Ins2k C i^i Ins3k `'kD)"k_V)
4	3	eleq2i 2417	. . . 4 |- (<<A, B>> e. (C o.k D) <-> <<A, B>> e. (( Ins2k C i^i Ins3k `'kD)"k_V))
5		opkex 4113	. . . . 5 |- <<A, B>> e. _V
6	5	elimakv 4260	. . . 4 |- (<<A, B>> e. (( Ins2k C i^i Ins3k `'kD)"k_V) <-> E.y<<y, <<A, B>>>> e. ( Ins2k C i^i Ins3k `'kD))
7		vex 2862	. . . . . . . . . 10 |- y e. _V
8		opkelins2kg 4251	. . . . . . . . . 10 |- ((y e. _V /\ <<A, B>> e. _V) -> (<<y, <<A, B>>>> e. Ins2k C <-> E.xE.zE.w(y = {{x}} /\ <<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
9	7, 5, 8	mp2an 653	. . . . . . . . 9 |- (<<y, <<A, B>>>> e. Ins2k C <-> E.xE.zE.w(y = {{x}} /\ <<A, B>> = <<z, w>> /\ <<x, w>> e. C))
10		3anass 938	. . . . . . . . . . . 12 |- ((y = {{x}} /\ <<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> (y = {{x}} /\ (<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
11	10	2exbii 1583	. . . . . . . . . . 11 |- (E.zE.w(y = {{x}} /\ <<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> E.zE.w(y = {{x}} /\ (<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
12		19.42vv 1907	. . . . . . . . . . 11 |- (E.zE.w(y = {{x}} /\ (<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) <-> (y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
13	11, 12	bitri 240	. . . . . . . . . 10 |- (E.zE.w(y = {{x}} /\ <<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> (y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
14	13	exbii 1582	. . . . . . . . 9 |- (E.xE.zE.w(y = {{x}} /\ <<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> E.x(y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
15	9, 14	bitri 240	. . . . . . . 8 |- (<<y, <<A, B>>>> e. Ins2k C <-> E.x(y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
16	15	anbi1i 676	. . . . . . 7 |- ((<<y, <<A, B>>>> e. Ins2k C /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> (E.x(y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD))
17		elin 3219	. . . . . . 7 |- (<<y, <<A, B>>>> e. ( Ins2k C i^i Ins3k `'kD) <-> (<<y, <<A, B>>>> e. Ins2k C /\ <<y, <<A, B>>>> e. Ins3k `'kD))
18		19.41v 1901	. . . . . . 7 |- (E.x((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> (E.x(y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD))
19	16, 17, 18	3bitr4i 268	. . . . . 6 |- (<<y, <<A, B>>>> e. ( Ins2k C i^i Ins3k `'kD) <-> E.x((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD))
20	19	exbii 1582	. . . . 5 |- (E.y<<y, <<A, B>>>> e. ( Ins2k C i^i Ins3k `'kD) <-> E.yE.x((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD))
21		excom 1741	. . . . 5 |- (E.yE.x((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> E.xE.y((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD))
22		anass 630	. . . . . . . 8 |- (((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> (y = {{x}} /\ (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<y, <<A, B>>>> e. Ins3k `'kD)))
23	22	exbii 1582	. . . . . . 7 |- (E.y((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> E.y(y = {{x}} /\ (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<y, <<A, B>>>> e. Ins3k `'kD)))
24		snex 4111	. . . . . . . 8 |- {{x}} e. _V
25		opkeq1 4059	. . . . . . . . . 10 |- (y = {{x}} -> <<y, <<A, B>>>> = <<{{x}}, <<A, B>>>>)
26	25	eleq1d 2419	. . . . . . . . 9 |- (y = {{x}} -> (<<y, <<A, B>>>> e. Ins3k `'kD <-> <<{{x}}, <<A, B>>>> e. Ins3k `'kD))
27	26	anbi2d 684	. . . . . . . 8 |- (y = {{x}} -> ((E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD)))
28	24, 27	ceqsexv 2894	. . . . . . 7 |- (E.y(y = {{x}} /\ (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<y, <<A, B>>>> e. Ins3k `'kD)) <-> (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD))
29	23, 28	bitri 240	. . . . . 6 |- (E.y((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD))
30	29	exbii 1582	. . . . 5 |- (E.xE.y((y = {{x}} /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) /\ <<y, <<A, B>>>> e. Ins3k `'kD) <-> E.x(E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD))
31	20, 21, 30	3bitri 262	. . . 4 |- (E.y<<y, <<A, B>>>> e. ( Ins2k C i^i Ins3k `'kD) <-> E.x(E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD))
32	4, 6, 31	3bitri 262	. . 3 |- (<<A, B>> e. (C o.k D) <-> E.x(E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD))
33		ancom 437	. . . . 5 |- ((E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD) <-> (<<{{x}}, <<A, B>>>> e. Ins3k `'kD /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)))
34		vex 2862	. . . . . . . 8 |- x e. _V
35		otkelins3kg 4254	. . . . . . . 8 |- ((x e. _V /\ A e. _V /\ B e. _V) -> (<<{{x}}, <<A, B>>>> e. Ins3k `'kD <-> <<x, A>> e. `'kD))
36	34, 35	mp3an1 1264	. . . . . . 7 |- ((A e. _V /\ B e. _V) -> (<<{{x}}, <<A, B>>>> e. Ins3k `'kD <-> <<x, A>> e. `'kD))
37		opkelcnvkg 4249	. . . . . . . . 9 |- ((x e. _V /\ A e. _V) -> (<<x, A>> e. `'kD <-> <<A, x>> e. D))
38	34, 37	mpan 651	. . . . . . . 8 |- (A e. _V -> (<<x, A>> e. `'kD <-> <<A, x>> e. D))
39	38	adantr 451	. . . . . . 7 |- ((A e. _V /\ B e. _V) -> (<<x, A>> e. `'kD <-> <<A, x>> e. D))
40	36, 39	bitrd 244	. . . . . 6 |- ((A e. _V /\ B e. _V) -> (<<{{x}}, <<A, B>>>> e. Ins3k `'kD <-> <<A, x>> e. D))
41		eqcom 2355	. . . . . . . . . 10 |- (<<A, B>> = <<z, w>> <-> <<z, w>> = <<A, B>>)
42	41	anbi1i 676	. . . . . . . . 9 |- ((<<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> (<<z, w>> = <<A, B>> /\ <<x, w>> e. C))
43	42	2exbii 1583	. . . . . . . 8 |- (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> E.zE.w(<<z, w>> = <<A, B>> /\ <<x, w>> e. C))
44		vex 2862	. . . . . . . . . . . . . 14 |- z e. _V
45		vex 2862	. . . . . . . . . . . . . 14 |- w e. _V
46		opkthg 4131	. . . . . . . . . . . . . 14 |- ((z e. _V /\ w e. _V /\ B e. _V) -> (<<z, w>> = <<A, B>> <-> (z = A /\ w = B)))
47	44, 45, 46	mp3an12 1267	. . . . . . . . . . . . 13 |- (B e. _V -> (<<z, w>> = <<A, B>> <-> (z = A /\ w = B)))
48	47	anbi1d 685	. . . . . . . . . . . 12 |- (B e. _V -> ((<<z, w>> = <<A, B>> /\ <<x, w>> e. C) <-> ((z = A /\ w = B) /\ <<x, w>> e. C)))
49		anass 630	. . . . . . . . . . . 12 |- (((z = A /\ w = B) /\ <<x, w>> e. C) <-> (z = A /\ (w = B /\ <<x, w>> e. C)))
50	48, 49	syl6bb 252	. . . . . . . . . . 11 |- (B e. _V -> ((<<z, w>> = <<A, B>> /\ <<x, w>> e. C) <-> (z = A /\ (w = B /\ <<x, w>> e. C))))
51	50	2exbidv 1628	. . . . . . . . . 10 |- (B e. _V -> (E.zE.w(<<z, w>> = <<A, B>> /\ <<x, w>> e. C) <-> E.zE.w(z = A /\ (w = B /\ <<x, w>> e. C))))
52	51	adantl 452	. . . . . . . . 9 |- ((A e. _V /\ B e. _V) -> (E.zE.w(<<z, w>> = <<A, B>> /\ <<x, w>> e. C) <-> E.zE.w(z = A /\ (w = B /\ <<x, w>> e. C))))
53		eeanv 1913	. . . . . . . . 9 |- (E.zE.w(z = A /\ (w = B /\ <<x, w>> e. C)) <-> (E.z z = A /\ E.w(w = B /\ <<x, w>> e. C)))
54	52, 53	syl6bb 252	. . . . . . . 8 |- ((A e. _V /\ B e. _V) -> (E.zE.w(<<z, w>> = <<A, B>> /\ <<x, w>> e. C) <-> (E.z z = A /\ E.w(w = B /\ <<x, w>> e. C))))
55	43, 54	syl5bb 248	. . . . . . 7 |- ((A e. _V /\ B e. _V) -> (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> (E.z z = A /\ E.w(w = B /\ <<x, w>> e. C))))
56		elisset 2869	. . . . . . . . . 10 |- (A e. _V -> E.z z = A)
57	56	biantrurd 494	. . . . . . . . 9 |- (A e. _V -> (E.w(w = B /\ <<x, w>> e. C) <-> (E.z z = A /\ E.w(w = B /\ <<x, w>> e. C))))
58	57	bicomd 192	. . . . . . . 8 |- (A e. _V -> ((E.z z = A /\ E.w(w = B /\ <<x, w>> e. C)) <-> E.w(w = B /\ <<x, w>> e. C)))
59		opkeq2 4060	. . . . . . . . . 10 |- (w = B -> <<x, w>> = <<x, B>>)
60	59	eleq1d 2419	. . . . . . . . 9 |- (w = B -> (<<x, w>> e. C <-> <<x, B>> e. C))
61	60	ceqsexgv 2971	. . . . . . . 8 |- (B e. _V -> (E.w(w = B /\ <<x, w>> e. C) <-> <<x, B>> e. C))
62	58, 61	sylan9bb 680	. . . . . . 7 |- ((A e. _V /\ B e. _V) -> ((E.z z = A /\ E.w(w = B /\ <<x, w>> e. C)) <-> <<x, B>> e. C))
63	55, 62	bitrd 244	. . . . . 6 |- ((A e. _V /\ B e. _V) -> (E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) <-> <<x, B>> e. C))
64	40, 63	anbi12d 691	. . . . 5 |- ((A e. _V /\ B e. _V) -> ((<<{{x}}, <<A, B>>>> e. Ins3k `'kD /\ E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C)) <-> (<<A, x>> e. D /\ <<x, B>> e. C)))
65	33, 64	syl5bb 248	. . . 4 |- ((A e. _V /\ B e. _V) -> ((E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD) <-> (<<A, x>> e. D /\ <<x, B>> e. C)))
66	65	exbidv 1626	. . 3 |- ((A e. _V /\ B e. _V) -> (E.x(E.zE.w(<<A, B>> = <<z, w>> /\ <<x, w>> e. C) /\ <<{{x}}, <<A, B>>>> e. Ins3k `'kD) <-> E.x(<<A, x>> e. D /\ <<x, B>> e. C)))
67	32, 66	syl5bb 248	. 2 |- ((A e. _V /\ B e. _V) -> (<<A, B>> e. (C o.k D) <-> E.x(<<A, x>> e. D /\ <<x, B>> e. C)))
68	1, 2, 67	syl2an 463	1 |- ((A e. V /\ B e. W) -> (<<A, B>> e. (C o.k D) <-> E.x(<<A, x>> e. D /\ <<x, B>> e. C)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859   i^i cin 3208  {csn 3737  <<copk 4057  `'_kccnvk 4175   Ins2_k cins2k 4176   Ins3_k cins3k 4177  "_kcimak 4179   o._k ccomk 4180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-3an 936  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-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-sn 3741  df-pr 3742  df-opk 4058  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190

This theorem is referenced by:  opkelcok  4262  opkelimagekg  4271