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Theorem sikeq 4241
 Description: Equality theorem for Kuratowski singleton image. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
sikeq (A = BSIk A = SIk B)

Proof of Theorem sikeq
Dummy variables x y z w t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2414 . . . . . . 7 (A = B → (⟪w, t A ↔ ⟪w, t B))
213anbi3d 1258 . . . . . 6 (A = B → ((y = {w} z = {t} w, t A) ↔ (y = {w} z = {t} w, t B)))
322exbidv 1628 . . . . 5 (A = B → (wt(y = {w} z = {t} w, t A) ↔ wt(y = {w} z = {t} w, t B)))
43anbi2d 684 . . . 4 (A = B → ((x = ⟪y, z wt(y = {w} z = {t} w, t A)) ↔ (x = ⟪y, z wt(y = {w} z = {t} w, t B))))
542exbidv 1628 . . 3 (A = B → (yz(x = ⟪y, z wt(y = {w} z = {t} w, t A)) ↔ yz(x = ⟪y, z wt(y = {w} z = {t} w, t B))))
65abbidv 2467 . 2 (A = B → {x yz(x = ⟪y, z wt(y = {w} z = {t} w, t A))} = {x yz(x = ⟪y, z wt(y = {w} z = {t} w, t B))})
7 df-sik 4192 . 2 SIk A = {x yz(x = ⟪y, z wt(y = {w} z = {t} w, t A))}
8 df-sik 4192 . 2 SIk B = {x yz(x = ⟪y, z wt(y = {w} z = {t} w, t B))}
96, 7, 83eqtr4g 2410 1 (A = BSIk A = SIk B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {csn 3737  ⟪copk 4057   SIk csik 4181 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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sik 4192 This theorem is referenced by:  sikeqi  4242  sikeqd  4243  imagekeq  4244  sikexg  4296
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