Theorem elimakvg 4259

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

Assertion

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

Distinct variable groups:   y,A   y,C

Allowed substitution hint:   V(y)

Proof of Theorem elimakvg

Step	Hyp	Ref	Expression
1		elimakg 4258	. 2 ⊢ (C ∈ V → (C ∈ (A “k V) ↔ ∃y ∈ V ⟪y, C⟫ ∈ A))
2		rexv 2874	. 2 ⊢ (∃y ∈ V ⟪y, C⟫ ∈ A ↔ ∃y⟪y, C⟫ ∈ A)
3	1, 2	syl6bb 252	1 ⊢ (C ∈ V → (C ∈ (A “k V) ↔ ∃y⟪y, C⟫ ∈ A))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∃wex 1541   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  ⟪copk 4058   “_k cimak 4180

This proof depends on 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 proof 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 used by:  elimakv  4261