Theorem elrfirn 42651

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 6754	. . 3 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2		elrfi 42650	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
3	1, 2	sylan2 592	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
4		imassrn 6100	. . . . . 6 ⊢ (𝐹 “ 𝑣) ⊆ ran 𝐹
5		pwexg 5396	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
6		ssexg 5341	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
7	1, 5, 6	syl2anr 596	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8		elpw2g 5351	. . . . . . 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 6750	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
13	12	ad2antlr 726	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14		inss2 4259	. . . . . . 7 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ Fin
15	14	sseli 4004	. . . . . 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 9381	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑣 ∈ Fin) → (𝐹 “ 𝑣) ∈ Fin)
18	13, 16, 17	syl2anc 583	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ Fin)
19	11, 18	elind 4223	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20		ffn 6747	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → 𝐹 Fn 𝐼)
21	20	ad2antlr 726	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22		inss1 4258	. . . . . . . 8 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
23	22	sseli 4004	. . . . . . 7 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
24	23	elpwid 4631	. . . . . 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 4259	. . . . . . 7 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
27	26	sseli 4004	. . . . . 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 9428	. . . . 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 2747	. . . . 5 ⊢ ((𝐹 “ 𝑣) = 𝑤 ↔ 𝑤 = (𝐹 “ 𝑣))
32	31	rexbii 3100	. . . 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 4973	. . . . . 6 ⊢ (𝑤 = (𝐹 “ 𝑣) → ∩ 𝑤 = ∩ (𝐹 “ 𝑣))
35	34	ineq2d 4241	. . . . 5 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐵 ∩ ∩ 𝑤) = (𝐵 ∩ ∩ (𝐹 “ 𝑣)))
36	35	eqeq2d 2751	. . . 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 5427	. 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 726	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40		inss1 4258	. . . . . . . . . 10 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
41	40	sseli 4004	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
42	41	elpwid 4631	. . . . . . . 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 42649	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
45	39, 43, 44	syl2anc 583	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
46	45	eqcomd 2746	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ (𝐹 “ 𝑣) = ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))
47	46	ineq2d 4241	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 ∩ ∩ (𝐹 “ 𝑣)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))
48	47	eqeq2d 2751	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
49	48	rexbidva 3183	. 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 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ∩ cint 4970  ∩ ciin 5016  ran crn 5701   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  Fincfn 9003  ficfi 9479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-dom 9005  df-fin 9007  df-fi 9480

This theorem is referenced by:  elrfirn2  42652