Theorem elimakv 4261

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

Hypothesis

Ref	Expression
elimak.1	⊢ C ∈ V

Assertion

Ref	Expression
elimakv	⊢ (C ∈ (A “k V) ↔ ∃y⟪y, C⟫ ∈ A)

Distinct variable groups:   y,A   y,C

Proof of Theorem elimakv

Step	Hyp	Ref	Expression
1		elimak.1	. 2 ⊢ C ∈ V
2		elimakvg 4259	. 2 ⊢ (C ∈ V → (C ∈ (A “k V) ↔ ∃y⟪y, C⟫ ∈ A))
3	1, 2	ax-mp 5	1 ⊢ (C ∈ (A “k V) ↔ ∃y⟪y, C⟫ ∈ A)

Syntax hints:   ↔ wb 176  ∃wex 1541   ∈ wcel 1710  Vcvv 2860  ⟪copk 4058   “_k cimak 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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059  df-imak 4190

This theorem is referenced by:  opkelcokg  4262  cokrelk  4285  dfpw12  4302  evenodddisjlem1  4516  dfswap2  4742