Theorem elp6 4264

Description: Membership in the P6 operator. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
elp6	⊢ (A ∈ V → (A ∈ P6 B ↔ ∀x⟪x, {A}⟫ ∈ B))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   V(x)

Proof of Theorem elp6

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3745	. . . . . 6 ⊢ (y = A → {y} = {A})
2	1	sneqd 3747	. . . . 5 ⊢ (y = A → {{y}} = {{A}})
3	2	xpkeq2d 4206	. . . 4 ⊢ (y = A → (V ×k {{y}}) = (V ×k {{A}}))
4	3	sseq1d 3299	. . 3 ⊢ (y = A → ((V ×k {{y}}) ⊆ B ↔ (V ×k {{A}}) ⊆ B))
5		df-p6 4192	. . 3 ⊢ P6 B = {y ∣ (V ×k {{y}}) ⊆ B}
6	4, 5	elab2g 2988	. 2 ⊢ (A ∈ V → (A ∈ P6 B ↔ (V ×k {{A}}) ⊆ B))
7		xpkssvvk 4211	. . . 4 ⊢ (V ×k {{A}}) ⊆ (V ×k V)
8		ssrelk 4212	. . . 4 ⊢ ((V ×k {{A}}) ⊆ (V ×k V) → ((V ×k {{A}}) ⊆ B ↔ ∀x∀y(⟪x, y⟫ ∈ (V ×k {{A}}) → ⟪x, y⟫ ∈ B)))
9	7, 8	ax-mp 5	. . 3 ⊢ ((V ×k {{A}}) ⊆ B ↔ ∀x∀y(⟪x, y⟫ ∈ (V ×k {{A}}) → ⟪x, y⟫ ∈ B))
10		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
11		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
12	10, 11	opkelxpk 4249	. . . . . . . 8 ⊢ (⟪x, y⟫ ∈ (V ×k {{A}}) ↔ (x ∈ V ∧ y ∈ {{A}}))
13	10	biantrur 492	. . . . . . . 8 ⊢ (y ∈ {{A}} ↔ (x ∈ V ∧ y ∈ {{A}}))
14		df-sn 3742	. . . . . . . . 9 ⊢ {{A}} = {y ∣ y = {A}}
15	14	abeq2i 2461	. . . . . . . 8 ⊢ (y ∈ {{A}} ↔ y = {A})
16	12, 13, 15	3bitr2i 264	. . . . . . 7 ⊢ (⟪x, y⟫ ∈ (V ×k {{A}}) ↔ y = {A})
17	16	imbi1i 315	. . . . . 6 ⊢ ((⟪x, y⟫ ∈ (V ×k {{A}}) → ⟪x, y⟫ ∈ B) ↔ (y = {A} → ⟪x, y⟫ ∈ B))
18	17	albii 1566	. . . . 5 ⊢ (∀y(⟪x, y⟫ ∈ (V ×k {{A}}) → ⟪x, y⟫ ∈ B) ↔ ∀y(y = {A} → ⟪x, y⟫ ∈ B))
19		snex 4112	. . . . . 6 ⊢ {A} ∈ V
20		opkeq2 4061	. . . . . . 7 ⊢ (y = {A} → ⟪x, y⟫ = ⟪x, {A}⟫)
21	20	eleq1d 2419	. . . . . 6 ⊢ (y = {A} → (⟪x, y⟫ ∈ B ↔ ⟪x, {A}⟫ ∈ B))
22	19, 21	ceqsalv 2886	. . . . 5 ⊢ (∀y(y = {A} → ⟪x, y⟫ ∈ B) ↔ ⟪x, {A}⟫ ∈ B)
23	18, 22	bitri 240	. . . 4 ⊢ (∀y(⟪x, y⟫ ∈ (V ×k {{A}}) → ⟪x, y⟫ ∈ B) ↔ ⟪x, {A}⟫ ∈ B)
24	23	albii 1566	. . 3 ⊢ (∀x∀y(⟪x, y⟫ ∈ (V ×k {{A}}) → ⟪x, y⟫ ∈ B) ↔ ∀x⟪x, {A}⟫ ∈ B)
25	9, 24	bitri 240	. 2 ⊢ ((V ×k {{A}}) ⊆ B ↔ ∀x⟪x, {A}⟫ ∈ B)
26	6, 25	syl6bb 252	1 ⊢ (A ∈ V → (A ∈ P6 B ↔ ∀x⟪x, {A}⟫ ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  {csn 3738  ⟪copk 4058   ×_k cxpk 4175   P6 cp6 4179

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-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-p6 4192

This theorem is referenced by:  p6exg  4291  dfuni12  4292  dfimak2  4299