MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fival Structured version   Visualization version   GIF version

Theorem fival 8475
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 elex 3364 . 2 (𝐴𝑉𝐴 ∈ V)
2 simpr 471 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
3 inss1 3982 . . . . . . . . . 10 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
43sseli 3749 . . . . . . . . 9 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
54elpwid 4310 . . . . . . . 8 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥𝐴)
6 eqvisset 3363 . . . . . . . . 9 (𝑦 = 𝑥 𝑥 ∈ V)
7 intex 4952 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
86, 7sylibr 224 . . . . . . . 8 (𝑦 = 𝑥𝑥 ≠ ∅)
9 intssuni2 4637 . . . . . . . 8 ((𝑥𝐴𝑥 ≠ ∅) → 𝑥 𝐴)
105, 8, 9syl2an 577 . . . . . . 7 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑥 𝐴)
112, 10eqsstrd 3789 . . . . . 6 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 𝐴)
12 selpw 4305 . . . . . 6 (𝑦 ∈ 𝒫 𝐴𝑦 𝐴)
1311, 12sylibr 224 . . . . 5 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑦 = 𝑥) → 𝑦 ∈ 𝒫 𝐴)
1413rexlimiva 3176 . . . 4 (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥𝑦 ∈ 𝒫 𝐴)
1514abssi 3827 . . 3 {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴
16 uniexg 7103 . . . 4 (𝐴𝑉 𝐴 ∈ V)
17 pwexg 4981 . . . 4 ( 𝐴 ∈ V → 𝒫 𝐴 ∈ V)
1816, 17syl 17 . . 3 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
19 ssexg 4939 . . 3 (({𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
2015, 18, 19sylancr 569 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V)
21 pweq 4301 . . . . . 6 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
2221ineq1d 3965 . . . . 5 (𝑧 = 𝐴 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
2322rexeqdv 3294 . . . 4 (𝑧 = 𝐴 → (∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥 ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥))
2423abbidv 2890 . . 3 (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥} = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
25 df-fi 8474 . . 3 fi = (𝑧 ∈ V ↦ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝑧 ∩ Fin)𝑦 = 𝑥})
2624, 25fvmptg 6423 . 2 ((𝐴 ∈ V ∧ {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥} ∈ V) → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
271, 20, 26syl2anc 567 1 (𝐴𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wne 2943  wrex 3062  Vcvv 3351  cin 3723  wss 3724  c0 4064  𝒫 cpw 4298   cuni 4575   cint 4612  cfv 6032  Fincfn 8110  ficfi 8473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040  df-fi 8474
This theorem is referenced by:  elfi  8476  fi0  8483
  Copyright terms: Public domain W3C validator