Theorem trfg 23386

Description: The trace operation and the filGen operation are inverses to one another in some sense, with filGen growing the base set and ↾_t shrinking it. See fgtr 23385 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
trfg	⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

Proof of Theorem trfg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		filfbas 23343	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ∈ (fBas‘𝐴))
2	1	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝐴))
3		filsspw 23346	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
4	3	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝐴)
5		simp2 1137	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ⊆ 𝑋)
6	5	sspwd 4614	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
7	4, 6	sstrd 3991	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝑋)
8		simp3 1138	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
9		fbasweak 23360	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
10	2, 7, 8, 9	syl3anc 1371	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
11		fgcl 23373	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
12	10, 11	syl 17	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
13		filtop 23350	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐴 ∈ 𝐹)
14	13	3ad2ant1 1133	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ∈ 𝐹)
15		restval 17368	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝑋filGen𝐹) ↾t 𝐴) = ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)))
16	12, 14, 15	syl2anc 584	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)))
17		elfg 23366	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐹) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)))
18	10, 17	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝑋filGen𝐹) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)))
19	18	simplbda 500	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)
20		simpll1 1212	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝐴))
21		simprl 769	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐹)
22		inss2 4228	. . . . . . . 8 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
23	22	a1i 11	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → (𝑥 ∩ 𝐴) ⊆ 𝐴)
24		simprr 771	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
25		filelss 23347	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
26	25	3ad2antl1 1185	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
27	26	ad2ant2r 745	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝐴)
28	24, 27	ssind 4231	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ (𝑥 ∩ 𝐴))
29		filss 23348	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ (𝑦 ∈ 𝐹 ∧ (𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝑦 ⊆ (𝑥 ∩ 𝐴))) → (𝑥 ∩ 𝐴) ∈ 𝐹)
30	20, 21, 23, 28, 29	syl13anc 1372	. . . . . 6 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
31	19, 30	rexlimddv 3161	. . . . 5 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
32	31	fmpttd 7111	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)):(𝑋filGen𝐹)⟶𝐹)
33	32	frnd 6722	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)) ⊆ 𝐹)
34	16, 33	eqsstrd 4019	. 2 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) ⊆ 𝐹)
35		filelss 23347	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
36	35	3ad2antl1 1185	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
37		df-ss 3964	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝑥 ∩ 𝐴) = 𝑥)
38	36, 37	sylib 217	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) = 𝑥)
39	12	adantr 481	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
40	14	adantr 481	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝐴 ∈ 𝐹)
41		ssfg 23367	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
42	10, 41	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ (𝑋filGen𝐹))
43	42	sselda 3981	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝑋filGen𝐹))
44		elrestr 17370	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
45	39, 40, 43, 44	syl3anc 1371	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
46	38, 45	eqeltrrd 2834	. 2 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ ((𝑋filGen𝐹) ↾t 𝐴))
47	34, 46	eqelssd 4002	1 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601   ↦ cmpt 5230  ran crn 5676  ‘cfv 6540  (class class class)co 7405   ↾_t crest 17362  fBascfbas 20924  filGencfg 20925  Filcfil 23340

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rest 17364  df-fbas 20933  df-fg 20934  df-fil 23341

This theorem is referenced by:  cmetss  24824  minveclem4a  24938