Theorem inficl 9421

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 9415	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
2		eqimss2 3967	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → 𝐴 ⊆ 𝑧)
3	2	biantrurd 542	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)))
4		eleq2 2849	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝑧 ↔ (𝑥 ∩ 𝑦) ∈ 𝐴))
5	4	raleqbi1dv 3329	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
6	5	raleqbi1dv 3329	. . . . . . 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 3629	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)} ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
9		intss1 4922	. . . . 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 9419	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)})
12	11	sseq1d 3939	. . . 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 3923	. . . 4 ⊢ ((fi‘𝐴) = 𝐴 ↔ ((fi‘𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ (fi‘𝐴)))
15	14	simplbi2com 508	. . 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 9418	. . . 4 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
18	17	rgen2 3202	. . 3 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴)
19		eleq2 2849	. . . . 5 ⊢ ((fi‘𝐴) = 𝐴 → ((𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ↔ (𝑥 ∩ 𝑦) ∈ 𝐴))
20	19	raleqbi1dv 3329	. . . 4 ⊢ ((fi‘𝐴) = 𝐴 → (∀𝑦 ∈ (fi‘𝐴)(𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴))
21	20	raleqbi1dv 3329	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089  ∩ cint 4906  ‘cfv 6539  ficfi 9406

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-om 7879  df-1o 8480  df-2o 8481  df-en 8978  df-fin 8981  df-fi 9407

This theorem is used by:  fipwuni  9422  fisn  9423  fitop  23224  ordtbaslem  23512  ptbasin2  23903  filfi  24184  fmfnfmlem3  24281  ustuqtop2  24567  ldgenpisys  34846