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Theorem inficl 8316
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 8310 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
2 eqimss2 3650 . . . . . . . 8 (𝑧 = 𝐴𝐴𝑧)
32biantrurd 529 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)))
4 eleq2 2688 . . . . . . . . 9 (𝑧 = 𝐴 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐴))
54raleqbi1dv 3141 . . . . . . . 8 (𝑧 = 𝐴 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
65raleqbi1dv 3141 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
73, 6bitr3d 270 . . . . . 6 (𝑧 = 𝐴 → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
87elabg 3345 . . . . 5 (𝐴𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
9 intss1 4483 . . . . 5 (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴)
108, 9syl6bir 244 . . . 4 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
11 dffi2 8314 . . . . 5 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
1211sseq1d 3624 . . . 4 (𝐴𝑉 → ((fi‘𝐴) ⊆ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
1310, 12sylibrd 249 . . 3 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14 eqss 3610 . . . 4 ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴𝐴 ⊆ (fi‘𝐴)))
1514simplbi2com 656 . . 3 (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
161, 13, 15sylsyld 61 . 2 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17 fiin 8313 . . . 4 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
1817rgen2a 2974 . . 3 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)
19 eleq2 2688 . . . . 5 ((fi‘𝐴) = 𝐴 → ((𝑥𝑦) ∈ (fi‘𝐴) ↔ (𝑥𝑦) ∈ 𝐴))
2019raleqbi1dv 3141 . . . 4 ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2120raleqbi1dv 3141 . . 3 ((fi‘𝐴) = 𝐴 → (∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2218, 21mpbii 223 . 2 ((fi‘𝐴) = 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
2316, 22impbid1 215 1 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  wral 2909  cin 3566  wss 3567   cint 4466  cfv 5876  ficfi 8301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-fin 7944  df-fi 8302
This theorem is referenced by:  fipwuni  8317  fisn  8318  fitop  20686  ordtbaslem  20973  ptbasin2  21362  filfi  21644  fmfnfmlem3  21741  ustuqtop2  22027  ldgenpisys  30203
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