Theorem elrfirn2 43707

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 5294	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐵 ↔ 𝐶 ⊆ 𝐵))
2	1	biimprd 251	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ⊆ 𝐵 → 𝐶 ∈ 𝒫 𝐵))
3	2	ralimdv 3176	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵))
4	3	imp 412	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵)
5		eqid 2760	. . . . 5 ⊢ (𝑦 ∈ 𝐼 ↦ 𝐶) = (𝑦 ∈ 𝐼 ↦ 𝐶)
6	5	fmpt 7110	. . . 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 43706	. . 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 603	. 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 4133	. . . . . 6 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
11	10	sseli 3904	. . . . 5 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
12	11	elpwid 4565	. . . 4 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
13		nffvmpt1 6896	. . . . . . . 8 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)
14		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑧((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
15		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦))
16	13, 14, 15	cbviin 4993	. . . . . . 7 ⊢ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
17		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝑦 ∈ 𝐼)
18		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
19		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
20	18, 19	ssexd 5285	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
21	5	fvmpt2 7005	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝐶 ∈ V) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
22	17, 20, 21	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
23	22	ex 418	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) → (𝐶 ⊆ 𝐵 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
24	23	ralimdva 3174	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
25	24	imp 412	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
26		ssralv 3977	. . . . . . . . 9 ⊢ (𝑣 ⊆ 𝐼 → (∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶 → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
27	25, 26	mpan9 516	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
28		iineq2 4971	. . . . . . . 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 2807	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 𝐶)
31	30	ineq2d 4117	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶))
32	31	eqeq2d 2771	. . . 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 605	. . 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 3184	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  𝒫 cpw 4556  {csn 4583  ∩ ciin 4951   ↦ cmpt 5185  ran crn 5651  ⟶wf 6534  ‘cfv 6538  Fincfn 8973  ficfi 9402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9403

This theorem is used by:  cmpfiiin  43708