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Theorem fiuni 9469
Description: The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
fiuni (𝐴𝑉 𝐴 = (fi‘𝐴))

Proof of Theorem fiuni
StepHypRef Expression
1 ssfii 9460 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
21unissd 4916 . 2 (𝐴𝑉 𝐴 (fi‘𝐴))
3 fipwuni 9467 . . . . 5 (fi‘𝐴) ⊆ 𝒫 𝐴
43unissi 4915 . . . 4 (fi‘𝐴) ⊆ 𝒫 𝐴
5 unipw 5454 . . . 4 𝒫 𝐴 = 𝐴
64, 5sseqtri 4031 . . 3 (fi‘𝐴) ⊆ 𝐴
76a1i 11 . 2 (𝐴𝑉 (fi‘𝐴) ⊆ 𝐴)
82, 7eqssd 4000 1 (𝐴𝑉 𝐴 = (fi‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wss 3950  𝒫 cpw 4599   cuni 4906  cfv 6560  ficfi 9451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-2o 8508  df-en 8987  df-fin 8990  df-fi 9452
This theorem is referenced by:  fipwss  9470  ordttopon  23202  ptbasfi  23590  xkouni  23608  alexsublem  24053  alexsub  24054  alexsubb  24055  alexsubALTlem3  24058  alexsubALTlem4  24059  ptcmplem1  24061  topjoin  36367
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