Theorem elrfirn 43159

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

Assertion

Ref	Expression
elrfirn	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))

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

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐼(𝑦)   𝑉(𝑦)

Proof of Theorem elrfirn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frn 6666	. . 3 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2		elrfi 43158	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
3	1, 2	sylan2 600	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
4		imassrn 6030	. . . . . 6 ⊢ (𝐹 “ 𝑣) ⊆ ran 𝐹
5		pwexg 5310	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
6		ssexg 5254	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
7	1, 5, 6	syl2anr 604	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8		elpw2g 5264	. . . . . . 7 ⊢ (ran 𝐹 ∈ V → ((𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹 ↔ (𝐹 “ 𝑣) ⊆ ran 𝐹))
9	7, 8	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → ((𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹 ↔ (𝐹 “ 𝑣) ⊆ ran 𝐹))
10	4, 9	mpbiri 260	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹)
11	10	adantr 482	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹)
12		ffun 6662	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
13	12	ad2antlr 734	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14		inss2 4169	. . . . . . 7 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ Fin
15	14	sseli 3913	. . . . . 6 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
16	15	adantl 483	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17		imafi 9219	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑣 ∈ Fin) → (𝐹 “ 𝑣) ∈ Fin)
18	13, 16, 17	syl2anc 591	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ Fin)
19	11, 18	elind 4132	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20		ffn 6659	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → 𝐹 Fn 𝐼)
21	20	ad2antlr 734	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22		inss1 4168	. . . . . . . 8 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
23	22	sseli 3913	. . . . . . 7 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
24	23	elpwid 4541	. . . . . 6 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ⊆ ran 𝐹)
25	24	adantl 483	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ⊆ ran 𝐹)
26		inss2 4169	. . . . . . 7 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
27	26	sseli 3913	. . . . . 6 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
28	27	adantl 483	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29		fipreima 9262	. . . . 5 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
30	21, 25, 28, 29	syl3anc 1380	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
31		eqcom 2748	. . . . 5 ⊢ ((𝐹 “ 𝑣) = 𝑤 ↔ 𝑤 = (𝐹 “ 𝑣))
32	31	rexbii 3088	. . . 4 ⊢ (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
33	30, 32	sylib 220	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
34		inteq 4883	. . . . . 6 ⊢ (𝑤 = (𝐹 “ 𝑣) → ∩ 𝑤 = ∩ (𝐹 “ 𝑣))
35	34	ineq2d 4152	. . . . 5 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐵 ∩ ∩ 𝑤) = (𝐵 ∩ ∩ (𝐹 “ 𝑣)))
36	35	eqeq2d 2752	. . . 4 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ 𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
37	36	adantl 483	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 = (𝐹 “ 𝑣)) → (𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ 𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
38	19, 33, 37	rexxfrd 5341	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
39	20	ad2antlr 734	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40		inss1 4168	. . . . . . . . . 10 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
41	40	sseli 3913	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
42	41	elpwid 4541	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
43	42	adantl 483	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ⊆ 𝐼)
44		imaiinfv 43157	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
45	39, 43, 44	syl2anc 591	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
46	45	eqcomd 2747	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ (𝐹 “ 𝑣) = ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))
47	46	ineq2d 4152	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 ∩ ∩ (𝐹 “ 𝑣)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))
48	47	eqeq2d 2752	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
49	48	rexbidva 3163	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
50	3, 38, 49	3bitrd 307	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  Vcvv 3433   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885  𝒫 cpw 4532  {csn 4558  ∩ cint 4880  ∩ ciin 4925  ran crn 5622   “ cima 5624  Fun wfun 6483   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  Fincfn 8887  ficfi 9317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-en 8888  df-dom 8889  df-fin 8891  df-fi 9318

This theorem is referenced by:  elrfirn2  43160