Theorem elrfirn2 37259

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 4827	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐵 ↔ 𝐶 ⊆ 𝐵))
2	1	biimprd 238	. . . . . 6 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ⊆ 𝐵 → 𝐶 ∈ 𝒫 𝐵))
3	2	ralimdv 2963	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵))
4	3	imp 445	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵)
5		eqid 2622	. . . . 5 ⊢ (𝑦 ∈ 𝐼 ↦ 𝐶) = (𝑦 ∈ 𝐼 ↦ 𝐶)
6	5	fmpt 6381	. . . 4 ⊢ (∀𝑦 ∈ 𝐼 𝐶 ∈ 𝒫 𝐵 ↔ (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵)
7	4, 6	sylib 208	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝑦 ∈ 𝐼 ↦ 𝐶):𝐼⟶𝒫 𝐵)
8		elrfirn 37258	. . 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 487	. 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 3833	. . . . . 6 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
11	10	sseli 3599	. . . . 5 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
12	11	elpwid 4170	. . . 4 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
13		nffvmpt1 6199	. . . . . . . 8 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)
14		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑧((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
15		fveq2 6191	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦))
16	13, 14, 15	cbviin 4558	. . . . . . 7 ⊢ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦)
17		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝑦 ∈ 𝐼)
18		simpll 790	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐵 ∈ 𝑉)
19		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ⊆ 𝐵)
20	18, 19	ssexd 4805	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ V)
21	5	fvmpt2 6291	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝐶 ∈ V) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
22	17, 20, 21	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) ∧ 𝐶 ⊆ 𝐵) → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
23	22	ex 450	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑦 ∈ 𝐼) → (𝐶 ⊆ 𝐵 → ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
24	23	ralimdva 2962	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → (∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵 → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
25	24	imp 445	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → ∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
26		ssralv 3666	. . . . . . . . 9 ⊢ (𝑣 ⊆ 𝐼 → (∀𝑦 ∈ 𝐼 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶 → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶))
27	25, 26	mpan9 486	. . . . . . . 8 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∀𝑦 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑦) = 𝐶)
28		iineq2 4538	. . . . . . . 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	syl5eq 2668	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧) = ∩ 𝑦 ∈ 𝑣 𝐶)
31	30	ineq2d 3814	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) ∧ 𝑣 ⊆ 𝐼) → (𝐵 ∩ ∩ 𝑧 ∈ 𝑣 ((𝑦 ∈ 𝐼 ↦ 𝐶)‘𝑧)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶))
32	31	eqeq2d 2632	. . . 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 491	. . 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 3049	. 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 268	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∩ ciin 4521   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888  Fincfn 7955  ficfi 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317

This theorem is referenced by:  cmpfiiin  37260