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Theorem fival 9404
Description: The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
fival (𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑉
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem fival
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-fi 9403 . 2 fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥})
2 pweq 4609 . . . . 5 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32ineq1d 4204 . . . 4 (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
43rexeqdv 3318 . . 3 (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥))
54abbidv 2793 . 2 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
6 elex 3485 . 2 (𝐴𝑉𝐴 ∈ V)
7 simpr 484 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
8 elinel1 4188 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4604 . . . . . . . 8 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥𝐴)
10 eqvisset 3484 . . . . . . . . 9 (𝑦 = 𝑥 𝑥 ∈ V)
11 intex 5328 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
1210, 11sylibr 233 . . . . . . . 8 (𝑦 = 𝑥𝑥 ≠ ∅)
13 intssuni2 4968 . . . . . . . 8 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
149, 12, 13syl2an 595 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑥 𝐴)
157, 14eqsstrd 4013 . . . . . 6 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 𝐴)
16 velpw 4600 . . . . . 6 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
1715, 16sylibr 233 . . . . 5 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 ∈ 𝒫 𝐴)
1817rexlimiva 3139 . . . 4 (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥𝑦 ∈ 𝒫 𝐴)
1918abssi 4060 . . 3 {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴
20 uniexg 7724 . . . 4 (𝐴𝑉 𝐴 ∈ V)
2120pwexd 5368 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
22 ssexg 5314 . . 3 (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
2319, 21, 22sylancr 586 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
241, 5, 6, 23fvmptd3 7012 1 (𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2701  wne 2932  wrex 3062  Vcvv 3466  cin 3940  wss 3941  c0 4315  𝒫 cpw 4595   cuni 4900   cint 4941  cfv 6534  Fincfn 8936  ficfi 9402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-fi 9403
This theorem is referenced by:  elfi  9405  fi0  9412
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