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Theorem inficl 8188
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 8182 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
2 eqimss2 3617 . . . . . . . 8 (𝑧 = 𝐴𝐴𝑧)
32biantrurd 527 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)))
4 eleq2 2673 . . . . . . . . 9 (𝑧 = 𝐴 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐴))
54raleqbi1dv 3119 . . . . . . . 8 (𝑧 = 𝐴 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
65raleqbi1dv 3119 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
73, 6bitr3d 268 . . . . . 6 (𝑧 = 𝐴 → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
87elabg 3316 . . . . 5 (𝐴𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
9 intss1 4418 . . . . 5 (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴)
108, 9syl6bir 242 . . . 4 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
11 dffi2 8186 . . . . 5 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
1211sseq1d 3591 . . . 4 (𝐴𝑉 → ((fi‘𝐴) ⊆ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
1310, 12sylibrd 247 . . 3 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14 eqss 3579 . . . 4 ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴𝐴 ⊆ (fi‘𝐴)))
1514simplbi2com 654 . . 3 (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
161, 13, 15sylsyld 58 . 2 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17 fiin 8185 . . . 4 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
1817rgen2a 2956 . . 3 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)
19 eleq2 2673 . . . . 5 ((fi‘𝐴) = 𝐴 → ((𝑥𝑦) ∈ (fi‘𝐴) ↔ (𝑥𝑦) ∈ 𝐴))
2019raleqbi1dv 3119 . . . 4 ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2120raleqbi1dv 3119 . . 3 ((fi‘𝐴) = 𝐴 → (∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2218, 21mpbii 221 . 2 ((fi‘𝐴) = 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
2316, 22impbid1 213 1 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {cab 2592  wral 2892  cin 3535  wss 3536   cint 4401  cfv 5787  ficfi 8173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-fin 7819  df-fi 8174
This theorem is referenced by:  fipwuni  8189  fisn  8190  fitop  20469  ordtbaslem  20741  ptbasin2  21130  filfi  21412  fmfnfmlem3  21509  ustuqtop2  21795  ldgenpisys  29359
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