Theorem opkelcok 4262

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 4261	. 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 2859  ⟪copk 4057   ∘_k ccomk 4180

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 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-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-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190

This theorem is referenced by:  imacok  4282  dfnnc2  4395  nnsucelrlem1  4424  dfop2lem1  4573  setconslem1  4731  setconslem2  4732  dfco1  4748