Theorem inficl 9422

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	⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 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.

Step	Hyp	Ref	Expression
1		ssfii 9416	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2		eqimss2 4041	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → 𝐴 ⊆ 𝑧)
3	2	biantrurd 533	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)))
4		eleq2 2822	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝑥 ∩ 𝑦) ∈ 𝐴))
5	4	raleqbi1dv 3333	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
6	5	raleqbi1dv 3333	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
7	3, 6	bitr3d 280	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
8	7	elabg 3666	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
9		intss1 4967	. . . . 5 ⊢ (𝐴 ∈ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} → ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ⊆ 𝐴)
10	8, 9	syl6bir 253	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ⊆ 𝐴))
11		dffi2 9420	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)})
12	11	sseq1d 4013	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((fi‘𝐴) ⊆ 𝐴 ↔ ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ⊆ 𝐴))
13	10, 12	sylibrd 258	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14		eqss 3997	. . . 4 ⊢ ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (fi‘𝐴)))
15	14	simplbi2com 503	. . 3 ⊢ (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
16	1, 13, 15	sylsyld 61	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17		fiin 9419	. . . 4 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
18	17	rgen2 3197	. . 3 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴)
19		eleq2 2822	. . . . 5 ⊢ ((fi‘𝐴) = 𝐴 → ((𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ↔ (𝑥 ∩ 𝑦) ∈ 𝐴))
20	19	raleqbi1dv 3333	. . . 4 ⊢ ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
21	20	raleqbi1dv 3333	. . 3 ⊢ ((fi‘𝐴) = 𝐴 → (∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
22	18, 21	mpbii 232	. 2 ⊢ ((fi‘𝐴) = 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)
23	16, 22	impbid1 224	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061   ∩ cin 3947   ⊆ wss 3948  ∩ cint 4950  ‘cfv 6543  ficfi 9407

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fi 9408

This theorem is referenced by:  fipwuni  9423  fisn  9424  fitop  22622  ordtbaslem  22912  ptbasin2  23302  filfi  23583  fmfnfmlem3  23680  ustuqtop2  23967  ldgenpisys  33450