Theorem dfuni3 4316

Description: Alternate definition of class union for existence proof. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
dfuni3	⊢ ∪A = ⋃1(◡k Sk “k A)

Proof of Theorem dfuni3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . 6 ⊢ y ∈ V
2		snex 4112	. . . . . 6 ⊢ {x} ∈ V
3	1, 2	opkelcnvk 4251	. . . . 5 ⊢ (⟪y, {x}⟫ ∈ ◡k Sk ↔ ⟪{x}, y⟫ ∈ Sk )
4		vex 2863	. . . . . 6 ⊢ x ∈ V
5	4, 1	elssetk 4271	. . . . 5 ⊢ (⟪{x}, y⟫ ∈ Sk ↔ x ∈ y)
6	3, 5	bitri 240	. . . 4 ⊢ (⟪y, {x}⟫ ∈ ◡k Sk ↔ x ∈ y)
7	6	rexbii 2640	. . 3 ⊢ (∃y ∈ A ⟪y, {x}⟫ ∈ ◡k Sk ↔ ∃y ∈ A x ∈ y)
8	4	eluni1 4174	. . . 4 ⊢ (x ∈ ⋃1(◡k Sk “k A) ↔ {x} ∈ (◡k Sk “k A))
9	2	elimak 4260	. . . 4 ⊢ ({x} ∈ (◡k Sk “k A) ↔ ∃y ∈ A ⟪y, {x}⟫ ∈ ◡k Sk )
10	8, 9	bitri 240	. . 3 ⊢ (x ∈ ⋃1(◡k Sk “k A) ↔ ∃y ∈ A ⟪y, {x}⟫ ∈ ◡k Sk )
11		eluni2 3896	. . 3 ⊢ (x ∈ ∪A ↔ ∃y ∈ A x ∈ y)
12	7, 10, 11	3bitr4ri 269	. 2 ⊢ (x ∈ ∪A ↔ x ∈ ⋃1(◡k Sk “k A))
13	12	eqriv 2350	1 ⊢ ∪A = ⋃1(◡k Sk “k A)

Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  {csn 3738  ∪cuni 3892  ⟪copk 4058  ⋃₁cuni1 4134  ^◡_kccnvk 4176   “_k cimak 4180   S_k cssetk 4184

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-opk 4059  df-1c 4137  df-uni1 4139  df-cnvk 4187  df-imak 4190  df-ssetk 4194

This theorem is referenced by:  uniexg  4317