Theorem dfuni3 4316

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

Assertion

Ref	Expression
dfuni3	|- U.A = ⋃1(`'k Sk "kA)

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 e. _V
2		snex 4112	. . . . . 6 |- {x} e. _V
3	1, 2	opkelcnvk 4251	. . . . 5 |- (<<y, {x}>> e. `'k Sk <-> <<{x}, y>> e. Sk )
4		vex 2863	. . . . . 6 |- x e. _V
5	4, 1	elssetk 4271	. . . . 5 |- (<<{x}, y>> e. Sk <-> x e. y)
6	3, 5	bitri 240	. . . 4 |- (<<y, {x}>> e. `'k Sk <-> x e. y)
7	6	rexbii 2640	. . 3 |- (E.y e. A <<y, {x}>> e. `'k Sk <-> E.y e. A x e. y)
8	4	eluni1 4174	. . . 4 |- (x e. ⋃1(`'k Sk "kA) <-> {x} e. (`'k Sk "kA))
9	2	elimak 4260	. . . 4 |- ({x} e. (`'k Sk "kA) <-> E.y e. A <<y, {x}>> e. `'k Sk )
10	8, 9	bitri 240	. . 3 |- (x e. ⋃1(`'k Sk "kA) <-> E.y e. A <<y, {x}>> e. `'k Sk )
11		eluni2 3896	. . 3 |- (x e. U.A <-> E.y e. A x e. y)
12	7, 10, 11	3bitr4ri 269	. 2 |- (x e. U.A <-> x e. ⋃1(`'k Sk "kA))
13	12	eqriv 2350	1 |- U.A = ⋃1(`'k Sk "kA)

Syntax hints:   = wceq 1642   e. wcel 1710  E.wrex 2616  {csn 3738  U.cuni 3892  <<copk 4058  ⋃₁cuni1 4134  `'_kccnvk 4176  "_kcimak 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