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Theorem imakeq1 4224
 Description: Equality theorem for Kuratowski image. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
imakeq1 (A = B → (Ak C) = (Bk C))

Proof of Theorem imakeq1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . 4 (A = B → (⟪y, x A ↔ ⟪y, x B))
21rexbidv 2635 . . 3 (A = B → (y Cy, x Ay Cy, x B))
32abbidv 2467 . 2 (A = B → {x y Cy, x A} = {x y Cy, x B})
4 df-imak 4189 . 2 (Ak C) = {x y Cy, x A}
5 df-imak 4189 . 2 (Bk C) = {x y Cy, x B}
63, 4, 53eqtr4g 2410 1 (A = B → (Ak C) = (Bk C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rex 2620  df-imak 4189 This theorem is referenced by:  imakeq1i  4226  imakeq1d  4228
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