Theorem dfidk2 4314

Description: Definition of I_k in terms of S_k. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
dfidk2	⊢ Ik = ( Sk ∩ ◡k Sk )

Proof of Theorem dfidk2

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

Step	Hyp	Ref	Expression
1		idkssvvk 4282	. 2 ⊢ Ik ⊆ (V ×k V)
2		inss1 3476	. . 3 ⊢ ( Sk ∩ ◡k Sk ) ⊆ Sk
3		ssetkssvvk 4279	. . 3 ⊢ Sk ⊆ (V ×k V)
4	2, 3	sstri 3282	. 2 ⊢ ( Sk ∩ ◡k Sk ) ⊆ (V ×k V)
5		eqss 3288	. . 3 ⊢ (x = y ↔ (x ⊆ y ∧ y ⊆ x))
6		vex 2863	. . . 4 ⊢ x ∈ V
7		vex 2863	. . . 4 ⊢ y ∈ V
8		opkelidkg 4275	. . . 4 ⊢ ((x ∈ V ∧ y ∈ V) → (⟪x, y⟫ ∈ Ik ↔ x = y))
9	6, 7, 8	mp2an 653	. . 3 ⊢ (⟪x, y⟫ ∈ Ik ↔ x = y)
10		elin 3220	. . . 4 ⊢ (⟪x, y⟫ ∈ ( Sk ∩ ◡k Sk ) ↔ (⟪x, y⟫ ∈ Sk ∧ ⟪x, y⟫ ∈ ◡k Sk ))
11		opkelssetkg 4269	. . . . . 6 ⊢ ((x ∈ V ∧ y ∈ V) → (⟪x, y⟫ ∈ Sk ↔ x ⊆ y))
12	6, 7, 11	mp2an 653	. . . . 5 ⊢ (⟪x, y⟫ ∈ Sk ↔ x ⊆ y)
13	6, 7	opkelcnvk 4251	. . . . . 6 ⊢ (⟪x, y⟫ ∈ ◡k Sk ↔ ⟪y, x⟫ ∈ Sk )
14		opkelssetkg 4269	. . . . . . 7 ⊢ ((y ∈ V ∧ x ∈ V) → (⟪y, x⟫ ∈ Sk ↔ y ⊆ x))
15	7, 6, 14	mp2an 653	. . . . . 6 ⊢ (⟪y, x⟫ ∈ Sk ↔ y ⊆ x)
16	13, 15	bitri 240	. . . . 5 ⊢ (⟪x, y⟫ ∈ ◡k Sk ↔ y ⊆ x)
17	12, 16	anbi12i 678	. . . 4 ⊢ ((⟪x, y⟫ ∈ Sk ∧ ⟪x, y⟫ ∈ ◡k Sk ) ↔ (x ⊆ y ∧ y ⊆ x))
18	10, 17	bitri 240	. . 3 ⊢ (⟪x, y⟫ ∈ ( Sk ∩ ◡k Sk ) ↔ (x ⊆ y ∧ y ⊆ x))
19	5, 9, 18	3bitr4i 268	. 2 ⊢ (⟪x, y⟫ ∈ Ik ↔ ⟪x, y⟫ ∈ ( Sk ∩ ◡k Sk ))
20	1, 4, 19	eqrelkriiv 4214	1 ⊢ Ik = ( Sk ∩ ◡k Sk )

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ⟪copk 4058   ×_k cxpk 4175  ^◡_kccnvk 4176   S_k cssetk 4184   I_k cidk 4185

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-ral 2620  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-cnvk 4187  df-ssetk 4194  df-idk 4196

This theorem is referenced by:  idkex  4315