Theorem fiin 9417

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 6930	. . . . . 6 ⊢ (𝐴 ∈ (fi‘𝐶) → 𝐶 ∈ V)
2		elfi 9408	. . . . . 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 267	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
5	4	adantr 482	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑥 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = ∩ 𝑥)
6		simpr 486	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → 𝐵 ∈ (fi‘𝐶))
7		elfi 9408	. . . . . 6 ⊢ ((𝐵 ∈ (fi‘𝐶) ∧ 𝐶 ∈ V) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
8	7	ancoms 460	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
9	1, 8	sylan 581	. . . 4 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐵 ∈ (fi‘𝐶) ↔ ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦))
10	6, 9	mpbid 231	. . 3 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦)
11		elin 3965	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑥 ∈ 𝒫 𝐶 ∧ 𝑥 ∈ Fin))
12		elin 3965	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ Fin))
13		pwuncl 7757	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐶 ∧ 𝑦 ∈ 𝒫 𝐶) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐶)
14		unfi 9172	. . . . . . . . . . 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 659	. . . . . . . . 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 3965	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin) ↔ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐶 ∧ (𝑥 ∪ 𝑦) ∈ Fin))
19	17, 18	sylibr 233	. . . . . . 7 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝑦 ∈ (𝒫 𝐶 ∩ Fin)) → (𝑥 ∪ 𝑦) ∈ (𝒫 𝐶 ∩ Fin))
20		ineq12 4208	. . . . . . . 8 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = (∩ 𝑥 ∩ ∩ 𝑦))
21		intun 4985	. . . . . . . 8 ⊢ ∩ (𝑥 ∪ 𝑦) = (∩ 𝑥 ∩ ∩ 𝑦)
22	20, 21	eqtr4di 2791	. . . . . . 7 ⊢ ((𝐴 = ∩ 𝑥 ∧ 𝐵 = ∩ 𝑦) → (𝐴 ∩ 𝐵) = ∩ (𝑥 ∪ 𝑦))
23		inteq 4954	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → ∩ 𝑧 = ∩ (𝑥 ∪ 𝑦))
24	23	rspceeqv 3634	. . . . . . 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 659	. . . . 5 ⊢ (((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) ∧ (𝑦 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐵 = ∩ 𝑦)) → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧)
27	26	rexlimdvaa 3157	. . . 4 ⊢ ((𝑥 ∈ (𝒫 𝐶 ∩ Fin) ∧ 𝐴 = ∩ 𝑥) → (∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)𝐵 = ∩ 𝑦 → ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
28	27	rexlimiva 3148	. . 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 5320	. . . 4 ⊢ (𝐴 ∈ (fi‘𝐶) → (𝐴 ∩ 𝐵) ∈ V)
31		elfi 9408	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
32	30, 1, 31	syl2anc 585	. . 3 ⊢ (𝐴 ∈ (fi‘𝐶) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
33	32	adantr 482	. 2 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → ((𝐴 ∩ 𝐵) ∈ (fi‘𝐶) ↔ ∃𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝐴 ∩ 𝐵) = ∩ 𝑧))
34	29, 33	mpbird 257	1 ⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  Vcvv 3475   ∪ cun 3947   ∩ cin 3948  𝒫 cpw 4603  ∩ cint 4951  ‘cfv 6544  Fincfn 8939  ficfi 9405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-en 8940  df-fin 8943  df-fi 9406

This theorem is referenced by:  dffi2  9418  inficl  9420  elfiun  9425  dffi3  9426  fibas  22480  ordtbas2  22695  fsubbas  23371