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Theorem dfuni12 4291
 Description: Alternate definition of unit union. (Contributed by SF, 15-Mar-2015.)
Assertion
Ref Expression
dfuni12 1A = P6 (V ×k A)

Proof of Theorem dfuni12
Dummy variables x z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.27v 1894 . . . 4 (z(z V {x} A) ↔ (z z V {x} A))
2 vex 2862 . . . . . 6 z V
3 snex 4111 . . . . . 6 {x} V
42, 3opkelxpk 4248 . . . . 5 (⟪z, {x}⟫ (V ×k A) ↔ (z V {x} A))
54albii 1566 . . . 4 (zz, {x}⟫ (V ×k A) ↔ z(z V {x} A))
62ax-gen 1546 . . . . 5 z z V
76biantrur 492 . . . 4 ({x} A ↔ (z z V {x} A))
81, 5, 73bitr4ri 269 . . 3 ({x} Azz, {x}⟫ (V ×k A))
9 vex 2862 . . . 4 x V
109eluni1 4173 . . 3 (x 1A ↔ {x} A)
11 elp6 4263 . . . 4 (x V → (x P6 (V ×k A) ↔ zz, {x}⟫ (V ×k A)))
129, 11ax-mp 8 . . 3 (x P6 (V ×k A) ↔ zz, {x}⟫ (V ×k A))
138, 10, 123bitr4i 268 . 2 (x 1Ax P6 (V ×k A))
1413eqriv 2350 1 1A = P6 (V ×k A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  ⟪copk 4057  ⋃1cuni1 4133   ×k cxpk 4174   P6 cp6 4178 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  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-uni1 4138  df-xpk 4185  df-p6 4191 This theorem is referenced by:  uni1exg  4292
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