Theorem elrfirn 37258

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 6053	. . 3 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → ran 𝐹 ⊆ 𝒫 𝐵)
2		elrfi 37257	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran 𝐹 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
3	1, 2	sylan2 491	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑤)))
4		imassrn 5477	. . . . . 6 ⊢ (𝐹 “ 𝑣) ⊆ ran 𝐹
5		pwexg 4850	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
6		ssexg 4804	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ∈ V) → ran 𝐹 ∈ V)
7	1, 5, 6	syl2anr 495	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → ran 𝐹 ∈ V)
8		elpw2g 4827	. . . . . . 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 248	. . . . 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 6048	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → Fun 𝐹)
13	12	ad2antlr 763	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → Fun 𝐹)
14		inss2 3834	. . . . . . 7 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ Fin
15	14	sseli 3599	. . . . . 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 8259	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑣 ∈ Fin) → (𝐹 “ 𝑣) ∈ Fin)
18	13, 16, 17	syl2anc 693	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ Fin)
19	11, 18	elind 3798	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐹 “ 𝑣) ∈ (𝒫 ran 𝐹 ∩ Fin))
20		ffn 6045	. . . . . 6 ⊢ (𝐹:𝐼⟶𝒫 𝐵 → 𝐹 Fn 𝐼)
21	20	ad2antlr 763	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → 𝐹 Fn 𝐼)
22		inss1 3833	. . . . . . . 8 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ 𝒫 ran 𝐹
23	22	sseli 3599	. . . . . . 7 ⊢ (𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin) → 𝑤 ∈ 𝒫 ran 𝐹)
24	23	elpwid 4170	. . . . . 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 3834	. . . . . . 7 ⊢ (𝒫 ran 𝐹 ∩ Fin) ⊆ Fin
27	26	sseli 3599	. . . . . 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 8272	. . . . 5 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑤 ⊆ ran 𝐹 ∧ 𝑤 ∈ Fin) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
30	21, 25, 28, 29	syl3anc 1326	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤)
31		eqcom 2629	. . . . 5 ⊢ ((𝐹 “ 𝑣) = 𝑤 ↔ 𝑤 = (𝐹 “ 𝑣))
32	31	rexbii 3041	. . . 4 ⊢ (∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)(𝐹 “ 𝑣) = 𝑤 ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
33	30, 32	sylib 208	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑤 ∈ (𝒫 ran 𝐹 ∩ Fin)) → ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝑤 = (𝐹 “ 𝑣))
34		inteq 4478	. . . . . 6 ⊢ (𝑤 = (𝐹 “ 𝑣) → ∩ 𝑤 = ∩ (𝐹 “ 𝑣))
35	34	ineq2d 3814	. . . . 5 ⊢ (𝑤 = (𝐹 “ 𝑣) → (𝐵 ∩ ∩ 𝑤) = (𝐵 ∩ ∩ (𝐹 “ 𝑣)))
36	35	eqeq2d 2632	. . . 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 4881	. 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 763	. . . . . . 7 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → 𝐹 Fn 𝐼)
40		inss1 3833	. . . . . . . . . 10 ⊢ (𝒫 𝐼 ∩ Fin) ⊆ 𝒫 𝐼
41	40	sseli 3599	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝐼 ∩ Fin) → 𝑣 ∈ 𝒫 𝐼)
42	41	elpwid 4170	. . . . . . . 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 37256	. . . . . . 7 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑣 ⊆ 𝐼) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
45	39, 43, 44	syl2anc 693	. . . . . 6 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦) = ∩ (𝐹 “ 𝑣))
46	45	eqcomd 2628	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → ∩ (𝐹 “ 𝑣) = ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))
47	46	ineq2d 3814	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐵 ∩ ∩ (𝐹 “ 𝑣)) = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))
48	47	eqeq2d 2632	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) ∧ 𝑣 ∈ (𝒫 𝐼 ∩ Fin)) → (𝐴 = (𝐵 ∩ ∩ (𝐹 “ 𝑣)) ↔ 𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦))))
49	48	rexbidva 3049	. 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 294	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∩ cint 4475  ∩ ciin 4521  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶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:  elrfirn2  37259