Theorem elrfirn2 42677

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 5283	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐵 ↔ 𝐶 ⊆ 𝐵))
2	1	biimprd 248	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ⊆ 𝐵 → 𝐶 ∈ 𝒫 𝐵))
3	2	ralimdv 3147	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵))
4	3	imp 406	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵)
5		eqid 2729	. . . . 5 ⊢ (𝑦 ∈ 𝐼 ↦ 𝐶) = (𝑦 ∈ 𝐼 ↦ 𝐶)
6	5	fmpt 7064	. . . 4 ⊢ (∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵 ↔ (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵)
7	4, 6	sylib 218	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵)
8		elrfirn 42676	. . 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 591	. 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 4196	. . . . . 6 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
11	10	sseli 3939	. . . . 5 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
12	11	elpwid 4568	. . . 4 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
13		nffvmpt1 6851	. . . . . . . 8 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)
14		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑧((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
15		fveq2 6840	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦))
16	13, 14, 15	cbviin 4996	. . . . . . 7 ⊢ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
17		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝑦 ∈ 𝐼)
18		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
19		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
20	18, 19	ssexd 5274	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
21	5	fvmpt2 6961	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝐶 ∈ V) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
22	17, 20, 21	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
23	22	ex 412	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) → (𝐶 ⊆ 𝐵 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
24	23	ralimdva 3145	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
25	24	imp 406	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
26		ssralv 4012	. . . . . . . . 9 ⊢ (𝑣 ⊆ 𝐼 → (∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶 → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
27	25, 26	mpan9 506	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
28		iineq2 4972	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶 → ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = ∩ 𝑦 ∈ 𝑣 𝐶)
29	27, 28	syl 17	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = ∩ 𝑦 ∈ 𝑣 𝐶)
30	16, 29	eqtrid 2776	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 𝐶)
31	30	ineq2d 4179	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶))
32	31	eqeq2d 2740	. . . 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 593	. . 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 3155	. 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 279	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ∩ ciin 4952   ↦ cmpt 5183  ran crn 5632  ⟶wf 6495  ‘cfv 6499  Fincfn 8895  ficfi 9337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-en 8896  df-dom 8897  df-fin 8899  df-fi 9338

This theorem is referenced by:  cmpfiiin  42678