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Theorem dffi2 9463
Description: The set of finite intersections is the smallest set that contains 𝐴 and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
dffi2 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝑉,𝑧
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem dffi2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 elex 3501 . 2 (𝐴𝑉𝐴 ∈ V)
2 vex 3484 . . . . . . . . . 10 𝑡 ∈ V
3 elfi 9453 . . . . . . . . . 10 ((𝑡 ∈ V ∧ 𝐴 ∈ V) → (𝑡 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑡 = 𝑥))
42, 3mpan 690 . . . . . . . . 9 (𝐴 ∈ V → (𝑡 ∈ (fi‘𝐴) ↔ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑡 = 𝑥))
54biimpd 229 . . . . . . . 8 (𝐴 ∈ V → (𝑡 ∈ (fi‘𝐴) → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑡 = 𝑥))
6 df-rex 3071 . . . . . . . . 9 (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑡 = 𝑥 ↔ ∃𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥))
7 fiint 9366 . . . . . . . . . . . 12 (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝑧))
8 elinel1 4201 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4609 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥𝐴)
1093ad2ant2 1135 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑧𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑥𝐴)
11 simp1 1137 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑧𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝐴𝑧)
1210, 11sstrd 3994 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑧𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑥𝑧)
13 eqvisset 3500 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑥 𝑥 ∈ V)
14 intex 5344 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ≠ ∅ ↔ 𝑥 ∈ V)
1513, 14sylibr 234 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑥𝑥 ≠ ∅)
16153ad2ant3 1136 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑧𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑥 ≠ ∅)
17 elinel2 4202 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝒫 𝐴 ∩ Fin) → 𝑥 ∈ Fin)
18173ad2ant2 1135 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑧𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑥 ∈ Fin)
1912, 16, 183jca 1129 . . . . . . . . . . . . . . . . 17 ((𝐴𝑧𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → (𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin))
20193expib 1123 . . . . . . . . . . . . . . . 16 (𝐴𝑧 → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → (𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin)))
21 pm2.27 42 . . . . . . . . . . . . . . . 16 ((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → (((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝑧) → 𝑥𝑧))
2220, 21syl6 35 . . . . . . . . . . . . . . 15 (𝐴𝑧 → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → (((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝑧) → 𝑥𝑧)))
23 eleq1 2829 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑥 → (𝑡𝑧 𝑥𝑧))
2423biimprd 248 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑥 → ( 𝑥𝑧𝑡𝑧))
2524adantl 481 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → ( 𝑥𝑧𝑡𝑧))
2625a1i 11 . . . . . . . . . . . . . . 15 (𝐴𝑧 → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → ( 𝑥𝑧𝑡𝑧)))
2722, 26syldd 72 . . . . . . . . . . . . . 14 (𝐴𝑧 → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → (((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝑧) → 𝑡𝑧)))
2827com23 86 . . . . . . . . . . . . 13 (𝐴𝑧 → (((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝑧) → ((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧)))
2928alimdv 1916 . . . . . . . . . . . 12 (𝐴𝑧 → (∀𝑥((𝑥𝑧𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝑧) → ∀𝑥((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧)))
307, 29biimtrid 242 . . . . . . . . . . 11 (𝐴𝑧 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 → ∀𝑥((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧)))
3130imp 406 . . . . . . . . . 10 ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) → ∀𝑥((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧))
32 19.23v 1942 . . . . . . . . . 10 (∀𝑥((𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧) ↔ (∃𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧))
3331, 32sylib 218 . . . . . . . . 9 ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) → (∃𝑥(𝑥 ∈ (𝒫 𝐴 ∩ Fin) ∧ 𝑡 = 𝑥) → 𝑡𝑧))
346, 33biimtrid 242 . . . . . . . 8 ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) → (∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑡 = 𝑥𝑡𝑧))
355, 34sylan9 507 . . . . . . 7 ((𝐴 ∈ V ∧ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)) → (𝑡 ∈ (fi‘𝐴) → 𝑡𝑧))
3635ssrdv 3989 . . . . . 6 ((𝐴 ∈ V ∧ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)) → (fi‘𝐴) ⊆ 𝑧)
3736ex 412 . . . . 5 (𝐴 ∈ V → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) → (fi‘𝐴) ⊆ 𝑧))
3837alrimiv 1927 . . . 4 (𝐴 ∈ V → ∀𝑧((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) → (fi‘𝐴) ⊆ 𝑧))
39 ssintab 4965 . . . 4 ((fi‘𝐴) ⊆ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ ∀𝑧((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) → (fi‘𝐴) ⊆ 𝑧))
4038, 39sylibr 234 . . 3 (𝐴 ∈ V → (fi‘𝐴) ⊆ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
41 ssfii 9459 . . . . 5 (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
42 fiin 9462 . . . . . 6 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
4342rgen2 3199 . . . . 5 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)
44 fvex 6919 . . . . . 6 (fi‘𝐴) ∈ V
45 sseq2 4010 . . . . . . 7 (𝑧 = (fi‘𝐴) → (𝐴𝑧𝐴 ⊆ (fi‘𝐴)))
46 eleq2 2830 . . . . . . . . 9 (𝑧 = (fi‘𝐴) → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ (fi‘𝐴)))
4746raleqbi1dv 3338 . . . . . . . 8 (𝑧 = (fi‘𝐴) → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)))
4847raleqbi1dv 3338 . . . . . . 7 (𝑧 = (fi‘𝐴) → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)))
4945, 48anbi12d 632 . . . . . 6 (𝑧 = (fi‘𝐴) → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ (𝐴 ⊆ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴))))
5044, 49elab 3679 . . . . 5 ((fi‘𝐴) ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ (𝐴 ⊆ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)))
5141, 43, 50sylanblrc 590 . . . 4 (𝐴 ∈ V → (fi‘𝐴) ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
52 intss1 4963 . . . 4 ((fi‘𝐴) ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ (fi‘𝐴))
5351, 52syl 17 . . 3 (𝐴 ∈ V → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ (fi‘𝐴))
5440, 53eqssd 4001 . 2 (𝐴 ∈ V → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
551, 54syl 17 1 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   cint 4946  cfv 6561  Fincfn 8985  ficfi 9450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451
This theorem is referenced by:  fiss  9464  inficl  9465  dffi3  9471  fbssfi  23845
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