Theorem fiin 8211

Description: The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
fiin	⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))

Proof of Theorem fiin

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6131	. . . . . 6 ⊢ (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2		elfi 8202	. . . . . 6 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
3	1, 2	mpdan 699	. . . . 5 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
4	3	ibi 255	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
5	4	adantr 480	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
6		simpr 476	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7		elfi 8202	. . . . . 6 ⊢ ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
8	7	ancoms 468	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
9	1, 8	sylan 487	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
10	6, 9	mpbid 221	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦)
11		elin 3758	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin))
12		elin 3758	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin))
13		elpwi 4117	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝐶 → 𝑥 ⊆ 𝐶)
14		elpwi 4117	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝐶 → 𝑦 ⊆ 𝐶)
15	13, 14	anim12i 588	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐶))
16		unss 3749	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐶) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐶)
17	15, 16	sylib 207	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ⊆ 𝐶)
18		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
19		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
20	18, 19	unex 6854	. . . . . . . . . . . . 13 ⊢ (𝑥 ∪ 𝑦) ∈ V
21	20	elpw 4114	. . . . . . . . . . . 12 ⊢ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐶)
22	17, 21	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐶)
23		unfi 8112	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
24	22, 23	anim12i 588	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
25	24	an4s 865	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
26	11, 12, 25	syl2anb 495	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
27		elin 3758	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
28	26, 27	sylibr 223	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
29		ineq12 3771	. . . . . . . 8 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = (∩ 𝑥 ∩ ∩ 𝑦))
30		intun 4444	. . . . . . . 8 ⊢ ∩ (𝑥 ∪ 𝑦) = (∩ 𝑥 ∩ ∩ 𝑦)
31	29, 30	syl6eqr 2662	. . . . . . 7 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦))
32		inteq 4413	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ∩ 𝑧 = ∩ (𝑥 ∪ 𝑦))
33	32	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ((𝐴 ∩ 𝐵) = ∩ 𝑧 ↔ (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦)))
34	33	rspcev 3282	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
35	28, 31, 34	syl2an 493	. . . . . 6 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
36	35	an4s 865	. . . . 5 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
37	36	rexlimdvaa 3014	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
38	37	rexlimiva 3010	. . 3 ⊢ (∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥 → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
39	5, 10, 38	sylc 63	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
40		inex1g 4729	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∩ 𝐵) ∈ V)
41		elfi 8202	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
42	40, 1, 41	syl2anc 691	. . 3 ⊢ (𝐴 ∈ (fi‘𝐶) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
43	42	adantr 480	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
44	39, 43	mpbird 246	1 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∩ cint 4410  ‘cfv 5804  Fincfn 7841  ficfi 8199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200

This theorem is referenced by:  dffi2  8212  inficl  8214  elfiun  8219  dffi3  8220  fibas  20592  ordtbas2  20805  fsubbas  21481