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.

Step	Hyp	Ref	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
4	2, 3	opkelxpk 4248	. . . . 5 ⊢ (⟪z, {x}⟫ ∈ (V ×k A) ↔ (z ∈ V ∧ {x} ∈ A))
5	4	albii 1566	. . . 4 ⊢ (∀z⟪z, {x}⟫ ∈ (V ×k A) ↔ ∀z(z ∈ V ∧ {x} ∈ A))
6	2	ax-gen 1546	. . . . 5 ⊢ ∀z z ∈ V
7	6	biantrur 492	. . . 4 ⊢ ({x} ∈ A ↔ (∀z z ∈ V ∧ {x} ∈ A))
8	1, 5, 7	3bitr4ri 269	. . 3 ⊢ ({x} ∈ A ↔ ∀z⟪z, {x}⟫ ∈ (V ×k A))
9		vex 2862	. . . 4 ⊢ x ∈ V
10	9	eluni1 4173	. . 3 ⊢ (x ∈ ⋃1A ↔ {x} ∈ A)
11		elp6 4263	. . . 4 ⊢ (x ∈ V → (x ∈ P6 (V ×k A) ↔ ∀z⟪z, {x}⟫ ∈ (V ×k A)))
12	9, 11	ax-mp 5	. . 3 ⊢ (x ∈ P6 (V ×k A) ↔ ∀z⟪z, {x}⟫ ∈ (V ×k A))
13	8, 10, 12	3bitr4i 268	. 2 ⊢ (x ∈ ⋃1A ↔ x ∈ P6 (V ×k A))
14	13	eqriv 2350	1 ⊢ ⋃1A = P6 (V ×k A)

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-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  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