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Theorem elimakg 4257
 Description: Membership in a Kuratowski image. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
elimakg (C V → (C (Ak B) ↔ y By, 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.
StepHypRef Expression
1 opkeq2 4060 . . . 4 (x = C → ⟪y, x⟫ = ⟪y, C⟫)
21eleq1d 2419 . . 3 (x = C → (⟪y, x A ↔ ⟪y, C A))
32rexbidv 2635 . 2 (x = C → (y By, x Ay By, C A))
4 df-imak 4189 . 2 (Ak B) = {x y By, x A}
53, 4elab2g 2987 1 (C V → (C (Ak B) ↔ y By, C A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟪copk 4057   “k cimak 4179 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-opk 4058  df-imak 4189 This theorem is referenced by:  elimakvg  4258  elimak  4259
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