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Theorem fipwuni 9464
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 7759 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
21pwexd 5385 . . . 4 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3 pwuni 4950 . . . 4 𝐴 ⊆ 𝒫 𝐴
4 fiss 9462 . . . 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 4254 . . . . . . 7 (𝑥 𝐴 → (𝑥𝑦) ⊆ 𝐴)
7 vex 3482 . . . . . . . 8 𝑥 ∈ V
87elpw 4609 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
97inex1 5323 . . . . . . . 8 (𝑥𝑦) ∈ V
109elpw 4609 . . . . . . 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 3197 . . . 4 𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴(𝑥𝑦) ∈ 𝒫 𝐴
14 inficl 9463 . . . . 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 4036 . 2 (𝐴 ∈ V → (fi‘𝐴) ⊆ 𝒫 𝐴)
18 fvprc 6899 . . 3 𝐴 ∈ V → (fi‘𝐴) = ∅)
19 0ss 4406 . . 3 ∅ ⊆ 𝒫 𝐴
2018, 19eqsstrdi 4050 . 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 1537  wcel 2106  wral 3059  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  ficfi 9448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449
This theorem is referenced by:  fiuni  9466  ordtbas  23216
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