Theorem cokrelk 4284

Description: A Kuratowski composition is a Kuratowski relationship. (Contributed by SF, 4-Feb-2015.)

Assertion

Ref	Expression
cokrelk	|- (A o.k B) C_ (_V X.k _V)

Proof of Theorem cokrelk

Dummy variables x y a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cok 4190	. . . . 5 |- (A o.k B) = (( Ins2k A i^i Ins3k `'kB)"k_V)
2	1	eleq2i 2417	. . . 4 |- (x e. (A o.k B) <-> x e. (( Ins2k A i^i Ins3k `'kB)"k_V))
3		vex 2862	. . . . 5 |- x e. _V
4	3	elimakv 4260	. . . 4 |- (x e. (( Ins2k A i^i Ins3k `'kB)"k_V) <-> E.y<<y, x>> e. ( Ins2k A i^i Ins3k `'kB))
5	2, 4	bitri 240	. . 3 |- (x e. (A o.k B) <-> E.y<<y, x>> e. ( Ins2k A i^i Ins3k `'kB))
6		inss1 3475	. . . . . 6 |- ( Ins2k A i^i Ins3k `'kB) C_ Ins2k A
7	6	sseli 3269	. . . . 5 |- (<<y, x>> e. ( Ins2k A i^i Ins3k `'kB) -> <<y, x>> e. Ins2k A)
8		vex 2862	. . . . . . 7 |- y e. _V
9		opkelins2kg 4251	. . . . . . 7 |- ((y e. _V /\ x e. _V) -> (<<y, x>> e. Ins2k A <-> E.aE.bE.c(y = {{a}} /\ x = <<b, c>> /\ <<a, c>> e. A)))
10	8, 3, 9	mp2an 653	. . . . . 6 |- (<<y, x>> e. Ins2k A <-> E.aE.bE.c(y = {{a}} /\ x = <<b, c>> /\ <<a, c>> e. A))
11		vex 2862	. . . . . . . . . . 11 |- b e. _V
12		vex 2862	. . . . . . . . . . 11 |- c e. _V
13	11, 12	opkelxpk 4248	. . . . . . . . . . 11 |- (<<b, c>> e. (_V X.k _V) <-> (b e. _V /\ c e. _V))
14	11, 12, 13	mpbir2an 886	. . . . . . . . . 10 |- <<b, c>> e. (_V X.k _V)
15		eleq1 2413	. . . . . . . . . 10 |- (x = <<b, c>> -> (x e. (_V X.k _V) <-> <<b, c>> e. (_V X.k _V)))
16	14, 15	mpbiri 224	. . . . . . . . 9 |- (x = <<b, c>> -> x e. (_V X.k _V))
17	16	3ad2ant2 977	. . . . . . . 8 |- ((y = {{a}} /\ x = <<b, c>> /\ <<a, c>> e. A) -> x e. (_V X.k _V))
18	17	exlimiv 1634	. . . . . . 7 |- (E.c(y = {{a}} /\ x = <<b, c>> /\ <<a, c>> e. A) -> x e. (_V X.k _V))
19	18	exlimivv 1635	. . . . . 6 |- (E.aE.bE.c(y = {{a}} /\ x = <<b, c>> /\ <<a, c>> e. A) -> x e. (_V X.k _V))
20	10, 19	sylbi 187	. . . . 5 |- (<<y, x>> e. Ins2k A -> x e. (_V X.k _V))
21	7, 20	syl 15	. . . 4 |- (<<y, x>> e. ( Ins2k A i^i Ins3k `'kB) -> x e. (_V X.k _V))
22	21	exlimiv 1634	. . 3 |- (E.y<<y, x>> e. ( Ins2k A i^i Ins3k `'kB) -> x e. (_V X.k _V))
23	5, 22	sylbi 187	. 2 |- (x e. (A o.k B) -> x e. (_V X.k _V))
24	23	ssriv 3277	1 |- (A o.k B) C_ (_V X.k _V)

Syntax hints:   <-> wb 176   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859   i^i cin 3208   C_ wss 3257  {csn 3737  <<copk 4057   X._k cxpk 4174  `'_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-xpk 4185  df-ins2k 4187  df-imak 4189  df-cok 4190

This theorem is referenced by: (None)