Theorem opkelins2kg 4251

Description: Kuratowski ordered pair membership in Kuratowski insertion operator. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
opkelins2kg	⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ Ins2k C ↔ ∃x∃y∃z(A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C)))

Distinct variable groups:   x,A,y,z   x,B,y,z   x,C,y,z

Allowed substitution hints:   V(x,y,z)   W(x,y,z)

Proof of Theorem opkelins2kg

Dummy variables w t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ins2k 4187	. 2 ⊢ Ins2k C = {t ∣ ∃w∃u(t = ⟪w, u⟫ ∧ ∃x∃y∃z(w = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C))}
2		eqeq1 2359	. . . 4 ⊢ (w = A → (w = {{x}} ↔ A = {{x}}))
3	2	3anbi1d 1256	. . 3 ⊢ (w = A → ((w = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C) ↔ (A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C)))
4	3	3exbidv 1629	. 2 ⊢ (w = A → (∃x∃y∃z(w = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C) ↔ ∃x∃y∃z(A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C)))
5		eqeq1 2359	. . . 4 ⊢ (u = B → (u = ⟪y, z⟫ ↔ B = ⟪y, z⟫))
6	5	3anbi2d 1257	. . 3 ⊢ (u = B → ((A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C) ↔ (A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C)))
7	6	3exbidv 1629	. 2 ⊢ (u = B → (∃x∃y∃z(A = {{x}} ∧ u = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C) ↔ ∃x∃y∃z(A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C)))
8	1, 4, 7	opkelopkabg 4245	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟪A, B⟫ ∈ Ins2k C ↔ ∃x∃y∃z(A = {{x}} ∧ B = ⟪y, z⟫ ∧ ⟪x, z⟫ ∈ C)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3737  ⟪copk 4057   Ins2_k cins2k 4176

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-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-ins2k 4187

This theorem is referenced by:  otkelins2kg  4253  opkelcokg  4261  ins2kss  4279  cokrelk  4284