Theorem elrfirn2 43319

Description: Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
elrfirn2	⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶)))

Distinct variable groups:   𝑣,𝐴   𝑣,𝐵,𝑦   𝑣,𝐶   𝑣,𝐼,𝑦   𝑣,𝑉,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem elrfirn2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw2g 5304	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐵 ↔ 𝐶 ⊆ 𝐵))
2	1	biimprd 251	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ⊆ 𝐵 → 𝐶 ∈ 𝒫 𝐵))
3	2	ralimdv 3185	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵))
4	3	imp 411	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵)
5		eqid 2769	. . . . 5 ⊢ (𝑦 ∈ 𝐼 ↦ 𝐶) = (𝑦 ∈ 𝐼 ↦ 𝐶)
6	5	fmpt 7106	. . . 4 ⊢ (∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵 ↔ (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵)
7	4, 6	sylib 221	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵)
8		elrfirn 43318	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧))))
9	7, 8	syldan 602	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧))))
10		inss1 4197	. . . . . 6 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
11	10	sseli 3941	. . . . 5 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
12	11	elpwid 4576	. . . 4 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
13		nffvmpt1 6893	. . . . . . . 8 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)
14		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑧((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
15		fveq2 6882	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦))
16	13, 14, 15	cbviin 5004	. . . . . . 7 ⊢ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
17		simplr 780	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝑦 ∈ 𝐼)
18		simpll 778	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
19		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
20	18, 19	ssexd 5295	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
21	5	fvmpt2 7002	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝐶 ∈ V) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
22	17, 20, 21	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
23	22	ex 417	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) → (𝐶 ⊆ 𝐵 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
24	23	ralimdva 3183	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
25	24	imp 411	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
26		ssralv 4014	. . . . . . . . 9 ⊢ (𝑣 ⊆ 𝐼 → (∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶 → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
27	25, 26	mpan9 515	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
28		iineq2 4981	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶 → ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = ∩ 𝑦 ∈ 𝑣 𝐶)
29	27, 28	syl 18	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = ∩ 𝑦 ∈ 𝑣 𝐶)
30	16, 29	eqtrid 2816	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 𝐶)
31	30	ineq2d 4181	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶))
32	31	eqeq2d 2780	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → (𝐴 = (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶)))
33	12, 32	sylan2 604	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶)))
34	33	rexbidva 3193	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶)))
35	9, 34	bitrd 282	1 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  {csn 4594  ∩ ciin 4961   ↦ cmpt 5196  ran crn 5663  ⟶wf 6533  ‘cfv 6537  Fincfn 8943  ficfi 9370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863  df-1o 8453  df-en 8944  df-dom 8945  df-fin 8947  df-fi 9371

This theorem is referenced by:  cmpfiiin  43320