Theorem opkelcok 4263

Description: Membership in a Kuratowski composition. (Contributed by SF, 13-Jan-2015.)

Hypotheses

Ref	Expression
opkelcok.1	⊢ A ∈ V
opkelcok.2	⊢ B ∈ V

Assertion

Ref	Expression
opkelcok	⊢ (⟪A, B⟫ ∈ (C ∘k D) ↔ ∃x(⟪A, x⟫ ∈ D ∧ ⟪x, B⟫ ∈ C))

Distinct variable groups:   x,A   x,B   x,C   x,D

Proof of Theorem opkelcok

Step	Hyp	Ref	Expression
1		opkelcok.1	. 2 ⊢ A ∈ V
2		opkelcok.2	. 2 ⊢ B ∈ V
3		opkelcokg 4262	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (⟪A, B⟫ ∈ (C ∘k D) ↔ ∃x(⟪A, x⟫ ∈ D ∧ ⟪x, B⟫ ∈ C)))
4	1, 2, 3	mp2an 653	1 ⊢ (⟪A, B⟫ ∈ (C ∘k D) ↔ ∃x(⟪A, x⟫ ∈ D ∧ ⟪x, B⟫ ∈ C))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  Vcvv 2860  ⟪copk 4058   ∘_k ccomk 4181

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-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-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191

This theorem is referenced by:  imacok  4283  dfnnc2  4396  nnsucelrlem1  4425  dfop2lem1  4574  setconslem1  4732  setconslem2  4733  dfco1  4749