Theorem elrfirn 42798

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 6658	. . 3 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2		elrfi 42797	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
3	1, 2	sylan2 593	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
4		imassrn 6019	. . . . . 6 ⊢ (𝐹 “ 𝑣) ⊆ ran 𝐹
5		pwexg 5314	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
6		ssexg 5259	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
7	1, 5, 6	syl2anr 597	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8		elpw2g 5269	. . . . . . 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 258	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹)
11	10	adantr 480	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹)
12		ffun 6654	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
13	12	ad2antlr 727	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14		inss2 4185	. . . . . . 7 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ Fin
15	14	sseli 3925	. . . . . 6 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
16	15	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17		imafi 9199	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑣 ∈ Fin) → (𝐹 “ 𝑣) ∈ Fin)
18	13, 16, 17	syl2anc 584	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ Fin)
19	11, 18	elind 4147	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20		ffn 6651	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → 𝐹 Fn 𝐼)
21	20	ad2antlr 727	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22		inss1 4184	. . . . . . . 8 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
23	22	sseli 3925	. . . . . . 7 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
24	23	elpwid 4556	. . . . . 6 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ⊆ ran 𝐹)
25	24	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ⊆ ran 𝐹)
26		inss2 4185	. . . . . . 7 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
27	26	sseli 3925	. . . . . 6 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
28	27	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29		fipreima 9242	. . . . 5 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
30	21, 25, 28, 29	syl3anc 1373	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
31		eqcom 2738	. . . . 5 ⊢ ((𝐹 “ 𝑣) = 𝑤 ↔ 𝑤 = (𝐹 “ 𝑣))
32	31	rexbii 3079	. . . 4 ⊢ (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
33	30, 32	sylib 218	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
34		inteq 4898	. . . . . 6 ⊢ (𝑤 = (𝐹 “ 𝑣) → ∩ 𝑤 = ∩ (𝐹 “ 𝑣))
35	34	ineq2d 4167	. . . . 5 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐵 ∩ ∩ 𝑤) = (𝐵 ∩ ∩ (𝐹 “ 𝑣)))
36	35	eqeq2d 2742	. . . 4 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ 𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
37	36	adantl 481	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 = (𝐹 “ 𝑣)) → (𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ 𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
38	19, 33, 37	rexxfrd 5345	. 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 727	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40		inss1 4184	. . . . . . . . . 10 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
41	40	sseli 3925	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
42	41	elpwid 4556	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
43	42	adantl 481	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ⊆ 𝐼)
44		imaiinfv 42796	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
45	39, 43, 44	syl2anc 584	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
46	45	eqcomd 2737	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ (𝐹 “ 𝑣) = ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))
47	46	ineq2d 4167	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 ∩ ∩ (𝐹 “ 𝑣)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))
48	47	eqeq2d 2742	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
49	48	rexbidva 3154	. 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 305	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  {csn 4573  ∩ cint 4895  ∩ ciin 4940  ran crn 5615   “ cima 5617  Fun wfun 6475   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  Fincfn 8869  ficfi 9294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-en 8870  df-dom 8871  df-fin 8873  df-fi 9295

This theorem is referenced by:  elrfirn2  42799