Theorem elp6 4264

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

Assertion

Ref	Expression
elp6	|- (A e. V -> (A e. P6 B <-> A.x<<x, {A}>> e. 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 X.k {{y}}) = (_V X.k {{A}}))
4	3	sseq1d 3299	. . 3 |- (y = A -> ((_V X.k {{y}}) C_ B <-> (_V X.k {{A}}) C_ B))
5		df-p6 4192	. . 3 |- P6 B = {y | (_V X.k {{y}}) C_ B}
6	4, 5	elab2g 2988	. 2 |- (A e. V -> (A e. P6 B <-> (_V X.k {{A}}) C_ B))
7		xpkssvvk 4211	. . . 4 |- (_V X.k {{A}}) C_ (_V X.k _V)
8		ssrelk 4212	. . . 4 |- ((_V X.k {{A}}) C_ (_V X.k _V) -> ((_V X.k {{A}}) C_ B <-> A.xA.y(<<x, y>> e. (_V X.k {{A}}) -> <<x, y>> e. B)))
9	7, 8	ax-mp 5	. . 3 |- ((_V X.k {{A}}) C_ B <-> A.xA.y(<<x, y>> e. (_V X.k {{A}}) -> <<x, y>> e. B))
10		vex 2863	. . . . . . . . 9 |- x e. _V
11		vex 2863	. . . . . . . . 9 |- y e. _V
12	10, 11	opkelxpk 4249	. . . . . . . 8 |- (<<x, y>> e. (_V X.k {{A}}) <-> (x e. _V /\ y e. {{A}}))
13	10	biantrur 492	. . . . . . . 8 |- (y e. {{A}} <-> (x e. _V /\ y e. {{A}}))
14		df-sn 3742	. . . . . . . . 9 |- {{A}} = {y | y = {A}}
15	14	abeq2i 2461	. . . . . . . 8 |- (y e. {{A}} <-> y = {A})
16	12, 13, 15	3bitr2i 264	. . . . . . 7 |- (<<x, y>> e. (_V X.k {{A}}) <-> y = {A})
17	16	imbi1i 315	. . . . . 6 |- ((<<x, y>> e. (_V X.k {{A}}) -> <<x, y>> e. B) <-> (y = {A} -> <<x, y>> e. B))
18	17	albii 1566	. . . . 5 |- (A.y(<<x, y>> e. (_V X.k {{A}}) -> <<x, y>> e. B) <-> A.y(y = {A} -> <<x, y>> e. B))
19		snex 4112	. . . . . 6 |- {A} e. _V
20		opkeq2 4061	. . . . . . 7 |- (y = {A} -> <<x, y>> = <<x, {A}>>)
21	20	eleq1d 2419	. . . . . 6 |- (y = {A} -> (<<x, y>> e. B <-> <<x, {A}>> e. B))
22	19, 21	ceqsalv 2886	. . . . 5 |- (A.y(y = {A} -> <<x, y>> e. B) <-> <<x, {A}>> e. B)
23	18, 22	bitri 240	. . . 4 |- (A.y(<<x, y>> e. (_V X.k {{A}}) -> <<x, y>> e. B) <-> <<x, {A}>> e. B)
24	23	albii 1566	. . 3 |- (A.xA.y(<<x, y>> e. (_V X.k {{A}}) -> <<x, y>> e. B) <-> A.x<<x, {A}>> e. B)
25	9, 24	bitri 240	. 2 |- ((_V X.k {{A}}) C_ B <-> A.x<<x, {A}>> e. B)
26	6, 25	syl6bb 252	1 |- (A e. V -> (A e. P6 B <-> A.x<<x, {A}>> e. B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2860   C_ wss 3258  {csn 3738  <<copk 4058   X._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