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Theorem inficl 8886
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 8880 . . 3 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
2 eqimss2 4010 . . . . . . . 8 (𝑧 = 𝐴𝐴𝑧)
32biantrurd 536 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)))
4 eleq2 2904 . . . . . . . . 9 (𝑧 = 𝐴 → ((𝑥𝑦) ∈ 𝑧 ↔ (𝑥𝑦) ∈ 𝐴))
54raleqbi1dv 3394 . . . . . . . 8 (𝑧 = 𝐴 → (∀𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
65raleqbi1dv 3394 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
73, 6bitr3d 284 . . . . . 6 (𝑧 = 𝐴 → ((𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
87elabg 3652 . . . . 5 (𝐴𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
9 intss1 4877 . . . . 5 (𝐴 ∈ {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} → {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴)
108, 9syl6bir 257 . . . 4 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
11 dffi2 8884 . . . . 5 (𝐴𝑉 → (fi‘𝐴) = {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)})
1211sseq1d 3984 . . . 4 (𝐴𝑉 → ((fi‘𝐴) ⊆ 𝐴 {𝑧 ∣ (𝐴𝑧 ∧ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ∈ 𝑧)} ⊆ 𝐴))
1310, 12sylibrd 262 . . 3 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14 eqss 3968 . . . 4 ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴𝐴 ⊆ (fi‘𝐴)))
1514simplbi2com 506 . . 3 (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
161, 13, 15sylsyld 61 . 2 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17 fiin 8883 . . . 4 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
1817rgen2 3198 . . 3 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴)
19 eleq2 2904 . . . . 5 ((fi‘𝐴) = 𝐴 → ((𝑥𝑦) ∈ (fi‘𝐴) ↔ (𝑥𝑦) ∈ 𝐴))
2019raleqbi1dv 3394 . . . 4 ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2120raleqbi1dv 3394 . . 3 ((fi‘𝐴) = 𝐴 → (∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥𝑦) ∈ (fi‘𝐴) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴))
2218, 21mpbii 236 . 2 ((fi‘𝐴) = 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴)
2316, 22impbid1 228 1 (𝐴𝑉 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3133  cin 3918  wss 3919   cint 4862  cfv 6343  ficfi 8871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-fin 8509  df-fi 8872
This theorem is referenced by:  fipwuni  8887  fisn  8888  fitop  21508  ordtbaslem  21796  ptbasin2  22186  filfi  22467  fmfnfmlem3  22564  ustuqtop2  22851  ldgenpisys  31482
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