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Theorem fival 8727
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 8726 . 2 fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥})
2 pweq 4460 . . . . 5 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32ineq1d 4112 . . . 4 (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
43rexeqdv 3376 . . 3 (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥))
54abbidv 2860 . 2 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
6 elex 3455 . 2 (𝐴𝑉𝐴 ∈ V)
7 simpr 485 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
8 elinel1 4097 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4469 . . . . . . . 8 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥𝐴)
10 eqvisset 3454 . . . . . . . . 9 (𝑦 = 𝑥 𝑥 ∈ V)
11 intex 5136 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
1210, 11sylibr 235 . . . . . . . 8 (𝑦 = 𝑥𝑥 ≠ ∅)
13 intssuni2 4811 . . . . . . . 8 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
149, 12, 13syl2an 595 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑥 𝐴)
157, 14eqsstrd 3930 . . . . . 6 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 𝐴)
16 selpw 4464 . . . . . 6 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
1715, 16sylibr 235 . . . . 5 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 ∈ 𝒫 𝐴)
1817rexlimiva 3244 . . . 4 (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥𝑦 ∈ 𝒫 𝐴)
1918abssi 3971 . . 3 {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴
20 uniexg 7330 . . . 4 (𝐴𝑉 𝐴 ∈ V)
2120pwexd 5176 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
22 ssexg 5123 . . 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 6662 1 (𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  {cab 2775  wne 2984  wrex 3106  Vcvv 3437  cin 3862  wss 3863  c0 4215  𝒫 cpw 4457   cuni 4749   cint 4786  cfv 6230  Fincfn 8362  ficfi 8725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-int 4787  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-iota 6194  df-fun 6232  df-fv 6238  df-fi 8726
This theorem is referenced by:  elfi  8728  fi0  8735
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