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Theorem fipwuni 9467
Description: The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
fipwuni (fi‘𝐴) ⊆ 𝒫 𝐴

Proof of Theorem fipwuni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniexg 7761 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
21pwexd 5378 . . . 4 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3 pwuni 4944 . . . 4 𝐴 ⊆ 𝒫 𝐴
4 fiss 9465 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐴) → (fi‘𝐴) ⊆ (fi‘𝒫 𝐴))
52, 3, 4sylancl 586 . . 3 (𝐴 ∈ V → (fi‘𝐴) ⊆ (fi‘𝒫 𝐴))
6 ssinss1 4245 . . . . . . 7 (𝑥 𝐴 → (𝑥𝑦) ⊆ 𝐴)
7 vex 3483 . . . . . . . 8 𝑥 ∈ V
87elpw 4603 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
97inex1 5316 . . . . . . . 8 (𝑥𝑦) ∈ V
109elpw 4603 . . . . . . 7 ((𝑥𝑦) ∈ 𝒫 𝐴 ↔ (𝑥𝑦) ⊆ 𝐴)
116, 8, 103imtr4i 292 . . . . . 6 (𝑥 ∈ 𝒫 𝐴 → (𝑥𝑦) ∈ 𝒫 𝐴)
1211adantr 480 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦) ∈ 𝒫 𝐴)
1312rgen2 3198 . . . 4 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
14 inficl 9466 . . . . 5 (𝒫 𝐴 ∈ V → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴 ↔ (fi‘𝒫 𝐴) = 𝒫 𝐴))
152, 14syl 17 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴 ↔ (fi‘𝒫 𝐴) = 𝒫 𝐴))
1613, 15mpbii 233 . . 3 (𝐴 ∈ V → (fi‘𝒫 𝐴) = 𝒫 𝐴)
175, 16sseqtrd 4019 . 2 (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝐴)
18 fvprc 6897 . . 3 𝐴 ∈ V → (fi‘𝐴) = ∅)
19 0ss 4399 . . 3 ∅ ⊆ 𝒫 𝐴
2018, 19eqsstrdi 4027 . 2 𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝐴)
2117, 20pm2.61i 182 1 (fi‘𝐴) ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cin 3949  wss 3950  c0 4332  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:  fiuni  9469  ordtbas  23201
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