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Theorem inficl 9463
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 9457 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
2 eqimss2 4055 . . . . . . . 8 (𝑧 = 𝐴𝐴𝑧)
32biantrurd 532 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)))
4 eleq2 2828 . . . . . . . . 9 (𝑧 = 𝐴 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐴))
54raleqbi1dv 3336 . . . . . . . 8 (𝑧 = 𝐴 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
65raleqbi1dv 3336 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
73, 6bitr3d 281 . . . . . 6 (𝑧 = 𝐴 → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
87elabg 3677 . . . . 5 (𝐴𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
9 intss1 4968 . . . . 5 (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴)
108, 9biimtrrdi 254 . . . 4 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
11 dffi2 9461 . . . . 5 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
1211sseq1d 4027 . . . 4 (𝐴𝑉 → ((fi‘𝐴) ⊆ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
1310, 12sylibrd 259 . . 3 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14 eqss 4011 . . . 4 ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴𝐴 ⊆ (fi‘𝐴)))
1514simplbi2com 502 . . 3 (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
161, 13, 15sylsyld 61 . 2 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17 fiin 9460 . . . 4 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
1817rgen2 3197 . . 3 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)
19 eleq2 2828 . . . . 5 ((fi‘𝐴) = 𝐴 → ((𝑥𝑦) ∈ (fi‘𝐴) ↔ (𝑥𝑦) ∈ 𝐴))
2019raleqbi1dv 3336 . . . 4 ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2120raleqbi1dv 3336 . . 3 ((fi‘𝐴) = 𝐴 → (∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2218, 21mpbii 233 . 2 ((fi‘𝐴) = 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
2316, 22impbid1 225 1 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  cin 3962  wss 3963   cint 4951  cfv 6563  ficfi 9448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449
This theorem is referenced by:  fipwuni  9464  fisn  9465  fitop  22922  ordtbaslem  23212  ptbasin2  23602  filfi  23883  fmfnfmlem3  23980  ustuqtop2  24267  ldgenpisys  34147
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