Theorem cokrelk 4285

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

Assertion

Ref	Expression
cokrelk	⊢ (A ∘k B) ⊆ (V ×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 4191	. . . . 5 ⊢ (A ∘k B) = (( Ins2k A ∩ Ins3k ◡kB) “k V)
2	1	eleq2i 2417	. . . 4 ⊢ (x ∈ (A ∘k B) ↔ x ∈ (( Ins2k A ∩ Ins3k ◡kB) “k V))
3		vex 2863	. . . . 5 ⊢ x ∈ V
4	3	elimakv 4261	. . . 4 ⊢ (x ∈ (( Ins2k A ∩ Ins3k ◡kB) “k V) ↔ ∃y⟪y, x⟫ ∈ ( Ins2k A ∩ Ins3k ◡kB))
5	2, 4	bitri 240	. . 3 ⊢ (x ∈ (A ∘k B) ↔ ∃y⟪y, x⟫ ∈ ( Ins2k A ∩ Ins3k ◡kB))
6		inss1 3476	. . . . . 6 ⊢ ( Ins2k A ∩ Ins3k ◡kB) ⊆ Ins2k A
7	6	sseli 3270	. . . . 5 ⊢ (⟪y, x⟫ ∈ ( Ins2k A ∩ Ins3k ◡kB) → ⟪y, x⟫ ∈ Ins2k A)
8		vex 2863	. . . . . . 7 ⊢ y ∈ V
9		opkelins2kg 4252	. . . . . . 7 ⊢ ((y ∈ V ∧ x ∈ V) → (⟪y, x⟫ ∈ Ins2k A ↔ ∃a∃b∃c(y = {{a}} ∧ x = ⟪b, c⟫ ∧ ⟪a, c⟫ ∈ A)))
10	8, 3, 9	mp2an 653	. . . . . 6 ⊢ (⟪y, x⟫ ∈ Ins2k A ↔ ∃a∃b∃c(y = {{a}} ∧ x = ⟪b, c⟫ ∧ ⟪a, c⟫ ∈ A))
11		vex 2863	. . . . . . . . . . 11 ⊢ b ∈ V
12		vex 2863	. . . . . . . . . . 11 ⊢ c ∈ V
13	11, 12	opkelxpk 4249	. . . . . . . . . . 11 ⊢ (⟪b, c⟫ ∈ (V ×k V) ↔ (b ∈ V ∧ c ∈ V))
14	11, 12, 13	mpbir2an 886	. . . . . . . . . 10 ⊢ ⟪b, c⟫ ∈ (V ×k V)
15		eleq1 2413	. . . . . . . . . 10 ⊢ (x = ⟪b, c⟫ → (x ∈ (V ×k V) ↔ ⟪b, c⟫ ∈ (V ×k V)))
16	14, 15	mpbiri 224	. . . . . . . . 9 ⊢ (x = ⟪b, c⟫ → x ∈ (V ×k V))
17	16	3ad2ant2 977	. . . . . . . 8 ⊢ ((y = {{a}} ∧ x = ⟪b, c⟫ ∧ ⟪a, c⟫ ∈ A) → x ∈ (V ×k V))
18	17	exlimiv 1634	. . . . . . 7 ⊢ (∃c(y = {{a}} ∧ x = ⟪b, c⟫ ∧ ⟪a, c⟫ ∈ A) → x ∈ (V ×k V))
19	18	exlimivv 1635	. . . . . 6 ⊢ (∃a∃b∃c(y = {{a}} ∧ x = ⟪b, c⟫ ∧ ⟪a, c⟫ ∈ A) → x ∈ (V ×k V))
20	10, 19	sylbi 187	. . . . 5 ⊢ (⟪y, x⟫ ∈ Ins2k A → x ∈ (V ×k V))
21	7, 20	syl 15	. . . 4 ⊢ (⟪y, x⟫ ∈ ( Ins2k A ∩ Ins3k ◡kB) → x ∈ (V ×k V))
22	21	exlimiv 1634	. . 3 ⊢ (∃y⟪y, x⟫ ∈ ( Ins2k A ∩ Ins3k ◡kB) → x ∈ (V ×k V))
23	5, 22	sylbi 187	. 2 ⊢ (x ∈ (A ∘k B) → x ∈ (V ×k V))
24	23	ssriv 3278	1 ⊢ (A ∘k B) ⊆ (V ×k V)

Syntax hints:   ↔ wb 176   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  {csn 3738  ⟪copk 4058   ×_k cxpk 4175  ^◡_kccnvk 4176   Ins2_k cins2k 4177   Ins3_k cins3k 4178   “_k cimak 4180   ∘_k ccomk 4181

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059  df-xpk 4186  df-ins2k 4188  df-imak 4190  df-cok 4191

This theorem is referenced by: (None)