Theorem fiin 8328

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 6221	. . . . . 6 ⊢ (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2		elfi 8319	. . . . . 6 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
3	1, 2	mpdan 702	. . . . 5 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
4	3	ibi 256	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
5	4	adantr 481	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
6		simpr 477	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7		elfi 8319	. . . . . 6 ⊢ ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
8	7	ancoms 469	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
9	1, 8	sylan 488	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
10	6, 9	mpbid 222	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦)
11		elin 3796	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin))
12		elin 3796	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin))
13		elpwi 4168	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝒫 𝐶 → 𝑥 ⊆ 𝐶)
14		elpwi 4168	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝐶 → 𝑦 ⊆ 𝐶)
15	13, 14	anim12i 590	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐶))
16		unss 3787	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐶 ∧ 𝑦 ⊆ 𝐶) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐶)
17	15, 16	sylib 208	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ⊆ 𝐶)
18		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
19		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
20	18, 19	unex 6956	. . . . . . . . . . . . 13 ⊢ (𝑥 ∪ 𝑦) ∈ V
21	20	elpw 4164	. . . . . . . . . . . 12 ⊢ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐶)
22	17, 21	sylibr 224	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐶)
23		unfi 8227	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
24	22, 23	anim12i 590	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
25	24	an4s 869	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
26	11, 12, 25	syl2anb 496	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
27		elin 3796	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
28	26, 27	sylibr 224	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
29		ineq12 3809	. . . . . . . 8 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = (∩ 𝑥 ∩ ∩ 𝑦))
30		intun 4509	. . . . . . . 8 ⊢ ∩ (𝑥 ∪ 𝑦) = (∩ 𝑥 ∩ ∩ 𝑦)
31	29, 30	syl6eqr 2674	. . . . . . 7 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦))
32		inteq 4478	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ∩ 𝑧 = ∩ (𝑥 ∪ 𝑦))
33	32	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ((𝐴 ∩ 𝐵) = ∩ 𝑧 ↔ (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦)))
34	33	rspcev 3309	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
35	28, 31, 34	syl2an 494	. . . . . 6 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
36	35	an4s 869	. . . . 5 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
37	36	rexlimdvaa 3032	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
38	37	rexlimiva 3028	. . 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 65	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
40		inex1g 4801	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∩ 𝐵) ∈ V)
41		elfi 8319	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
42	40, 1, 41	syl2anc 693	. . 3 ⊢ (𝐴 ∈ (fi‘𝐶) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
43	42	adantr 481	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
44	39, 43	mpbird 247	1 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∩ cint 4475  ‘cfv 5888  Fincfn 7955  ficfi 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317

This theorem is referenced by:  dffi2  8329  inficl  8331  elfiun  8336  dffi3  8337  fibas  20781  ordtbas2  20995  fsubbas  21671