Theorem dfuni12 4291

Description: Alternate definition of unit union. (Contributed by SF, 15-Mar-2015.)

Assertion

Ref	Expression
dfuni12	|- ⋃1A = P6 (_V X.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 |- (A.z(z e. _V /\ {x} e. A) <-> (A.z z e. _V /\ {x} e. A))
2		vex 2862	. . . . . 6 |- z e. _V
3		snex 4111	. . . . . 6 |- {x} e. _V
4	2, 3	opkelxpk 4248	. . . . 5 |- (<<z, {x}>> e. (_V X.k A) <-> (z e. _V /\ {x} e. A))
5	4	albii 1566	. . . 4 |- (A.z<<z, {x}>> e. (_V X.k A) <-> A.z(z e. _V /\ {x} e. A))
6	2	ax-gen 1546	. . . . 5 |- A.z z e. _V
7	6	biantrur 492	. . . 4 |- ({x} e. A <-> (A.z z e. _V /\ {x} e. A))
8	1, 5, 7	3bitr4ri 269	. . 3 |- ({x} e. A <-> A.z<<z, {x}>> e. (_V X.k A))
9		vex 2862	. . . 4 |- x e. _V
10	9	eluni1 4173	. . 3 |- (x e. ⋃1A <-> {x} e. A)
11		elp6 4263	. . . 4 |- (x e. _V -> (x e. P6 (_V X.k A) <-> A.z<<z, {x}>> e. (_V X.k A)))
12	9, 11	ax-mp 8	. . 3 |- (x e. P6 (_V X.k A) <-> A.z<<z, {x}>> e. (_V X.k A))
13	8, 10, 12	3bitr4i 268	. 2 |- (x e. ⋃1A <-> x e. P6 (_V X.k A))
14	13	eqriv 2350	1 |- ⋃1A = P6 (_V X.k A)

Syntax hints:   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737  <<copk 4057  ⋃₁cuni1 4133   X._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