Theorem inficl 9386

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 9380	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2		eqimss2 3997	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → 𝐴 ⊆ 𝑧)
3	2	biantrurd 541	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)))
4		eleq2 2852	. . . . . . . . 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 284	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
8	7	elabg 3636	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
9		intss1 4929	. . . . 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	biimtrrdi 257	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ⊆ 𝐴))
11		dffi2 9384	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)})
12	11	sseq1d 3969	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((fi‘𝐴) ⊆ 𝐴 ↔ ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ⊆ 𝐴))
13	10, 12	sylibrd 262	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → (fi‘𝐴) ⊆ 𝐴))
14		eqss 3953	. . . 4 ⊢ ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (fi‘𝐴)))
15	14	simplbi2com 507	. . 3 ⊢ (𝐴 ⊆ (fi‘𝐴) → ((fi‘𝐴) ⊆ 𝐴 → (fi‘𝐴) = 𝐴))
16	1, 13, 15	sylsyld 62	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 → (fi‘𝐴) = 𝐴))
17		fiin 9383	. . . 4 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
18	17	rgen2 3205	. . 3 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴)
19		eleq2 2852	. . . . 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 236	. 2 ⊢ ((fi‘𝐴) = 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)
23	16, 22	impbid1 228	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079   ∩ cin 3905   ⊆ wss 3906  ∩ cint 4913  ‘cfv 6538  ficfi 9371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7864  df-1o 8454  df-2o 8455  df-en 8945  df-fin 8948  df-fi 9372

This theorem is referenced by:  fipwuni  9387  fisn  9388  fitop  23038  ordtbaslem  23326  ptbasin2  23716  filfi  23997  fmfnfmlem3  24094  ustuqtop2  24380  ldgenpisys  34537