Theorem fiin 8870

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 6678	. . . . . 6 ⊢ (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2		elfi 8861	. . . . . 6 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
3	1, 2	mpdan 686	. . . . 5 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∈ (fi‘𝐶) ↔ ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥))
4	3	ibi 270	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
5	4	adantr 484	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
6		simpr 488	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7		elfi 8861	. . . . . 6 ⊢ ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
8	7	ancoms 462	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
9	1, 8	sylan 583	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
10	6, 9	mpbid 235	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦)
11		elin 3897	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin))
12		elin 3897	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin))
13		pwuncl 7472	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐶)
14		unfi 8769	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin)
15	13, 14	anim12i 615	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
16	15	an4s 659	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin) ∧ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
17	11, 12, 16	syl2anb 600	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
18		elin 3897	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
19	17, 18	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
20		ineq12 4134	. . . . . . . 8 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = (∩ 𝑥 ∩ ∩ 𝑦))
21		intun 4870	. . . . . . . 8 ⊢ ∩ (𝑥 ∪ 𝑦) = (∩ 𝑥 ∩ ∩ 𝑦)
22	20, 21	eqtr4di 2851	. . . . . . 7 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦))
23		inteq 4841	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ∩ 𝑧 = ∩ (𝑥 ∪ 𝑦))
24	23	rspceeqv 3586	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
25	19, 22, 24	syl2an 598	. . . . . 6 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) ∧ (𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
26	25	an4s 659	. . . . 5 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
27	26	rexlimdvaa 3244	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
28	27	rexlimiva 3240	. . 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 5187	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∩ 𝐵) ∈ V)
31		elfi 8861	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
32	30, 1, 31	syl2anc 587	. . 3 ⊢ (𝐴 ∈ (fi‘𝐶) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
33	32	adantr 484	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
34	29, 33	mpbird 260	1 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ∩ cin 3880  𝒫 cpw 4497  ∩ cint 4838  ‘cfv 6324  Fincfn 8492  ficfi 8858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-en 8493  df-fin 8496  df-fi 8859

This theorem is referenced by:  dffi2  8871  inficl  8873  elfiun  8878  dffi3  8879  fibas  21582  ordtbas2  21796  fsubbas  22472