Theorem fiin 8880

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 6697	. . . . . 6 ⊢ (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2		elfi 8871	. . . . . 6 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
3	1, 2	mpdan 685	. . . . 5 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
4	3	ibi 269	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
5	4	adantr 483	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
6		simpr 487	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7		elfi 8871	. . . . . 6 ⊢ ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
8	7	ancoms 461	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
9	1, 8	sylan 582	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
10	6, 9	mpbid 234	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦)
11		elin 4168	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin))
12		elin 4168	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin))
13		pwuncl 7486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐶)
14		unfi 8779	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
15	13, 14	anim12i 614	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
16	15	an4s 658	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
17	11, 12, 16	syl2anb 599	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
18		elin 4168	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
19	17, 18	sylibr 236	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
20		ineq12 4183	. . . . . . . 8 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = (∩ 𝑥 ∩ ∩ 𝑦))
21		intun 4900	. . . . . . . 8 ⊢ ∩ (𝑥 ∪ 𝑦) = (∩ 𝑥 ∩ ∩ 𝑦)
22	20, 21	syl6eqr 2874	. . . . . . 7 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦))
23		inteq 4871	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ∩ 𝑧 = ∩ (𝑥 ∪ 𝑦))
24	23	rspceeqv 3637	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
25	19, 22, 24	syl2an 597	. . . . . 6 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
26	25	an4s 658	. . . . 5 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
27	26	rexlimdvaa 3285	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
28	27	rexlimiva 3281	. . 3 ⊢ (∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥 → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
29	5, 10, 28	sylc 65	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
30		inex1g 5215	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∩ 𝐵) ∈ V)
31		elfi 8871	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
32	30, 1, 31	syl2anc 586	. . 3 ⊢ (𝐴 ∈ (fi‘𝐶) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
33	32	adantr 483	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
34	29, 33	mpbird 259	1 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494   ∪ cun 3933   ∩ cin 3934  𝒫 cpw 4538  ∩ cint 4868  ‘cfv 6349  Fincfn 8503  ficfi 8868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-er 8283  df-en 8504  df-fin 8507  df-fi 8869

This theorem is referenced by:  dffi2  8881  inficl  8883  elfiun  8888  dffi3  8889  fibas  21579  ordtbas2  21793  fsubbas  22469