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Theorem uni1exg 4292
 Description: The unit union operator preserves sethood. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
uni1exg (A V → ⋃1A V)

Proof of Theorem uni1exg
StepHypRef Expression
1 dfuni12 4291 . 2 1A = P6 (V ×k A)
2 vvex 4109 . . . 4 V V
3 xpkexg 4288 . . . 4 ((V V A V) → (V ×k A) V)
42, 3mpan 651 . . 3 (A V → (V ×k A) V)
5 p6exg 4290 . . 3 ((V ×k A) V → P6 (V ×k A) V)
64, 5syl 15 . 2 (A VP6 (V ×k A) V)
71, 6syl5eqel 2437 1 (A V → ⋃1A V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  Vcvv 2859  ⋃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-xp 4079  ax-cnv 4080  ax-typlower 4086  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-cnvk 4186  df-p6 4191 This theorem is referenced by:  uni1ex  4293  uniexg  4316  intexg  4319  coexg  4749  siexg  4752
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