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Theorem fival 8860
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 8859 . 2 fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥})
2 pweq 4513 . . . . 5 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32ineq1d 4138 . . . 4 (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
43rexeqdv 3365 . . 3 (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥))
54abbidv 2862 . 2 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
6 elex 3459 . 2 (𝐴𝑉𝐴 ∈ V)
7 simpr 488 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
8 elinel1 4122 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4508 . . . . . . . 8 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥𝐴)
10 eqvisset 3458 . . . . . . . . 9 (𝑦 = 𝑥 𝑥 ∈ V)
11 intex 5204 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
1210, 11sylibr 237 . . . . . . . 8 (𝑦 = 𝑥𝑥 ≠ ∅)
13 intssuni2 4863 . . . . . . . 8 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
149, 12, 13syl2an 598 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑥 𝐴)
157, 14eqsstrd 3953 . . . . . 6 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 𝐴)
16 velpw 4502 . . . . . 6 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
1715, 16sylibr 237 . . . . 5 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 ∈ 𝒫 𝐴)
1817rexlimiva 3240 . . . 4 (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥𝑦 ∈ 𝒫 𝐴)
1918abssi 3997 . . 3 {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴
20 uniexg 7446 . . . 4 (𝐴𝑉 𝐴 ∈ V)
2120pwexd 5245 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
22 ssexg 5191 . . 3 (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
2319, 21, 22sylancr 590 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
241, 5, 6, 23fvmptd3 6768 1 (𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {cab 2776  wne 2987  wrex 3107  Vcvv 3441  cin 3880  wss 3881  c0 4243  𝒫 cpw 4497   cuni 4800   cint 4838  cfv 6324  Fincfn 8492  ficfi 8858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-fi 8859
This theorem is referenced by:  elfi  8861  fi0  8868
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