Theorem elrfirn 41004

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 6675	. . 3 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2		elrfi 41003	. . 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 6024	. . . . . 6 ⊢ (𝐹 “ 𝑣) ⊆ ran 𝐹
5		pwexg 5333	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
6		ssexg 5280	. . . . . . . 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 5301	. . . . . . 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 257	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹)
11	10	adantr 481	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ 𝒫 ran 𝐹)
12		ffun 6671	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
13	12	ad2antlr 725	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14		inss2 4189	. . . . . . 7 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ Fin
15	14	sseli 3940	. . . . . 6 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ Fin)
16	15	adantl 482	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ∈ Fin)
17		imafi 9119	. . . . 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 4154	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20		ffn 6668	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → 𝐹 Fn 𝐼)
21	20	ad2antlr 725	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22		inss1 4188	. . . . . . . 8 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
23	22	sseli 3940	. . . . . . 7 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
24	23	elpwid 4569	. . . . . 6 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ⊆ ran 𝐹)
25	24	adantl 482	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ⊆ ran 𝐹)
26		inss2 4189	. . . . . . 7 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
27	26	sseli 3940	. . . . . 6 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ Fin)
28	27	adantl 482	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝑤 ∈ Fin)
29		fipreima 9302	. . . . 5 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
30	21, 25, 28, 29	syl3anc 1371	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
31		eqcom 2743	. . . . 5 ⊢ ((𝐹 “ 𝑣) = 𝑤 ↔ 𝑤 = (𝐹 “ 𝑣))
32	31	rexbii 3097	. . . 4 ⊢ (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
33	30, 32	sylib 217	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
34		inteq 4910	. . . . . 6 ⊢ (𝑤 = (𝐹 “ 𝑣) → ∩ 𝑤 = ∩ (𝐹 “ 𝑣))
35	34	ineq2d 4172	. . . . 5 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐵 ∩ ∩ 𝑤) = (𝐵 ∩ ∩ (𝐹 “ 𝑣)))
36	35	eqeq2d 2747	. . . 4 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ 𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
37	36	adantl 482	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 = (𝐹 “ 𝑣)) → (𝐴 = (𝐵 ∩ ∩ 𝑤) ↔ 𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣))))
38	19, 33, 37	rexxfrd 5364	. 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 725	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40		inss1 4188	. . . . . . . . . 10 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
41	40	sseli 3940	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
42	41	elpwid 4569	. . . . . . . 8 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ⊆ 𝐼)
43	42	adantl 482	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝑣 ⊆ 𝐼)
44		imaiinfv 41002	. . . . . . 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 2742	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ (𝐹 “ 𝑣) = ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))
47	46	ineq2d 4172	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 ∩ ∩ (𝐹 “ 𝑣)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))
48	47	eqeq2d 2747	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
49	48	rexbidva 3173	. 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 304	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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  {csn 4586  ∩ cint 4907  ∩ ciin 4955  ran crn 5634   “ cima 5636  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  Fincfn 8883  ficfi 9346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-en 8884  df-fin 8887  df-fi 9347

This theorem is referenced by:  elrfirn2  41005