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Theorem fival 8864
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 8863 . 2 fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥})
2 pweq 4538 . . . . 5 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32ineq1d 4185 . . . 4 (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
43rexeqdv 3414 . . 3 (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥))
54abbidv 2882 . 2 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
6 elex 3510 . 2 (𝐴𝑉𝐴 ∈ V)
7 simpr 485 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
8 elinel1 4169 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4549 . . . . . . . 8 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥𝐴)
10 eqvisset 3509 . . . . . . . . 9 (𝑦 = 𝑥 𝑥 ∈ V)
11 intex 5231 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
1210, 11sylibr 235 . . . . . . . 8 (𝑦 = 𝑥𝑥 ≠ ∅)
13 intssuni2 4892 . . . . . . . 8 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
149, 12, 13syl2an 595 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑥 𝐴)
157, 14eqsstrd 4002 . . . . . 6 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 𝐴)
16 velpw 4543 . . . . . 6 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
1715, 16sylibr 235 . . . . 5 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 ∈ 𝒫 𝐴)
1817rexlimiva 3278 . . . 4 (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥𝑦 ∈ 𝒫 𝐴)
1918abssi 4043 . . 3 {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴
20 uniexg 7456 . . . 4 (𝐴𝑉 𝐴 ∈ V)
2120pwexd 5271 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
22 ssexg 5218 . . 3 (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
2319, 21, 22sylancr 587 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
241, 5, 6, 23fvmptd3 6783 1 (𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wrex 3136  Vcvv 3492  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   cint 4867  cfv 6348  Fincfn 8497  ficfi 8862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-fi 8863
This theorem is referenced by:  elfi  8865  fi0  8872
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