Theorem elimakg 4258

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

Assertion

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

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

Allowed substitution hint:   V(y)

Proof of Theorem elimakg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opkeq2 4061	. . . 4 ⊢ (x = C → ⟪y, x⟫ = ⟪y, C⟫)
2	1	eleq1d 2419	. . 3 ⊢ (x = C → (⟪y, x⟫ ∈ A ↔ ⟪y, C⟫ ∈ A))
3	2	rexbidv 2636	. 2 ⊢ (x = C → (∃y ∈ B ⟪y, x⟫ ∈ A ↔ ∃y ∈ B ⟪y, C⟫ ∈ A))
4		df-imak 4190	. 2 ⊢ (A “k B) = {x ∣ ∃y ∈ B ⟪y, x⟫ ∈ A}
5	3, 4	elab2g 2988	1 ⊢ (C ∈ V → (C ∈ (A “k B) ↔ ∃y ∈ B ⟪y, C⟫ ∈ A))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ⟪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:  elimakvg  4259  elimak  4260