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Theorem inficl 9455
Description: A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Assertion
Ref Expression
inficl (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝑉
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem inficl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssfii 9449 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
2 eqimss2 4036 . . . . . . . 8 (𝑧 = 𝐴𝐴𝑧)
32biantrurd 531 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)))
4 eleq2 2814 . . . . . . . . 9 (𝑧 = 𝐴 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐴))
54raleqbi1dv 3322 . . . . . . . 8 (𝑧 = 𝐴 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
65raleqbi1dv 3322 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
73, 6bitr3d 280 . . . . . 6 (𝑧 = 𝐴 → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
87elabg 3662 . . . . 5 (𝐴𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
9 intss1 4967 . . . . 5 (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴)
108, 9biimtrrdi 253 . . . 4 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
11 dffi2 9453 . . . . 5 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
1211sseq1d 4008 . . . 4 (𝐴𝑉 → ((fi‘𝐴) ⊆ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
1310, 12sylibrd 258 . . 3 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14 eqss 3992 . . . 4 ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴𝐴 ⊆ (fi‘𝐴)))
1514simplbi2com 501 . . 3 (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
161, 13, 15sylsyld 61 . 2 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17 fiin 9452 . . . 4 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
1817rgen2 3187 . . 3 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)
19 eleq2 2814 . . . . 5 ((fi‘𝐴) = 𝐴 → ((𝑥𝑦) ∈ (fi‘𝐴) ↔ (𝑥𝑦) ∈ 𝐴))
2019raleqbi1dv 3322 . . . 4 ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2120raleqbi1dv 3322 . . 3 ((fi‘𝐴) = 𝐴 → (∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2218, 21mpbii 232 . 2 ((fi‘𝐴) = 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
2316, 22impbid1 224 1 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2702  wral 3050  cin 3943  wss 3944   cint 4950  cfv 6549  ficfi 9440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-2o 8488  df-en 8965  df-fin 8968  df-fi 9441
This theorem is referenced by:  fipwuni  9456  fisn  9457  fitop  22851  ordtbaslem  23141  ptbasin2  23531  filfi  23812  fmfnfmlem3  23909  ustuqtop2  24196  ldgenpisys  33918
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