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Theorem elp6 4263
 Description: Membership in the P6 operator. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
elp6 (A V → (A P6 Bxx, {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.
StepHypRef Expression
1 sneq 3744 . . . . . 6 (y = A → {y} = {A})
21sneqd 3746 . . . . 5 (y = A → {{y}} = {{A}})
32xpkeq2d 4205 . . . 4 (y = A → (V ×k {{y}}) = (V ×k {{A}}))
43sseq1d 3298 . . 3 (y = A → ((V ×k {{y}}) B ↔ (V ×k {{A}}) B))
5 df-p6 4191 . . 3 P6 B = {y (V ×k {{y}}) B}
64, 5elab2g 2987 . 2 (A V → (A P6 B ↔ (V ×k {{A}}) B))
7 xpkssvvk 4210 . . . 4 (V ×k {{A}}) (V ×k V)
8 ssrelk 4211 . . . 4 ((V ×k {{A}}) (V ×k V) → ((V ×k {{A}}) Bxy(⟪x, y (V ×k {{A}}) → ⟪x, y B)))
97, 8ax-mp 8 . . 3 ((V ×k {{A}}) Bxy(⟪x, y (V ×k {{A}}) → ⟪x, y B))
10 vex 2862 . . . . . . . . 9 x V
11 vex 2862 . . . . . . . . 9 y V
1210, 11opkelxpk 4248 . . . . . . . 8 (⟪x, y (V ×k {{A}}) ↔ (x V y {{A}}))
1310biantrur 492 . . . . . . . 8 (y {{A}} ↔ (x V y {{A}}))
14 df-sn 3741 . . . . . . . . 9 {{A}} = {y y = {A}}
1514abeq2i 2460 . . . . . . . 8 (y {{A}} ↔ y = {A})
1612, 13, 153bitr2i 264 . . . . . . 7 (⟪x, y (V ×k {{A}}) ↔ y = {A})
1716imbi1i 315 . . . . . 6 ((⟪x, y (V ×k {{A}}) → ⟪x, y B) ↔ (y = {A} → ⟪x, y B))
1817albii 1566 . . . . 5 (y(⟪x, y (V ×k {{A}}) → ⟪x, y B) ↔ y(y = {A} → ⟪x, y B))
19 snex 4111 . . . . . 6 {A} V
20 opkeq2 4060 . . . . . . 7 (y = {A} → ⟪x, y⟫ = ⟪x, {A}⟫)
2120eleq1d 2419 . . . . . 6 (y = {A} → (⟪x, y B ↔ ⟪x, {A}⟫ B))
2219, 21ceqsalv 2885 . . . . 5 (y(y = {A} → ⟪x, y B) ↔ ⟪x, {A}⟫ B)
2318, 22bitri 240 . . . 4 (y(⟪x, y (V ×k {{A}}) → ⟪x, y B) ↔ ⟪x, {A}⟫ B)
2423albii 1566 . . 3 (xy(⟪x, y (V ×k {{A}}) → ⟪x, y B) ↔ xx, {A}⟫ B)
259, 24bitri 240 . 2 ((V ×k {{A}}) Bxx, {A}⟫ B)
266, 25syl6bb 252 1 (A V → (A P6 Bxx, {A}⟫ B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  {csn 3737  ⟪copk 4057   ×k cxpk 4174   P6 cp6 4178 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ral 2619  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-p6 4191 This theorem is referenced by:  p6exg  4290  dfuni12  4291  dfimak2  4298
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