MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fiuni Structured version   Visualization version   GIF version

Theorem fiuni 8501
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 8492 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
21unissd 4614 . 2 (𝐴𝑉 𝐴 (fi‘𝐴))
3 fipwuni 8499 . . . . 5 (fi‘𝐴) ⊆ 𝒫 𝐴
43unissi 4613 . . . 4 (fi‘𝐴) ⊆ 𝒫 𝐴
5 unipw 5067 . . . 4 𝒫 𝐴 = 𝐴
64, 5sseqtri 3778 . . 3 (fi‘𝐴) ⊆ 𝐴
76a1i 11 . 2 (𝐴𝑉 (fi‘𝐴) ⊆ 𝐴)
82, 7eqssd 3761 1 (𝐴𝑉 𝐴 = (fi‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wss 3715  𝒫 cpw 4302   cuni 4588  cfv 6049  ficfi 8483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-fin 8127  df-fi 8484
This theorem is referenced by:  fipwss  8502  ordttopon  21219  ptbasfi  21606  xkouni  21624  alexsublem  22069  alexsub  22070  alexsubb  22071  alexsubALTlem3  22074  alexsubALTlem4  22075  ptcmplem1  22077  topjoin  32687
  Copyright terms: Public domain W3C validator