Theorem trfg 23915

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 23914 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 23872	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ∈ (fBas‘𝐴))
2	1	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝐴))
3		filsspw 23875	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
4	3	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝐴)
5		simp2 1136	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ⊆ 𝑋)
6	5	sspwd 4618	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
7	4, 6	sstrd 4006	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝑋)
8		simp3 1137	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
9		fbasweak 23889	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
10	2, 7, 8, 9	syl3anc 1370	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
11		fgcl 23902	. . . . 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 23879	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐴 ∈ 𝐹)
14	13	3ad2ant1 1132	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ∈ 𝐹)
15		restval 17473	. . . 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 23895	. . . . . . . 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 499	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)
20		simpll1 1211	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝐴))
21		simprl 771	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐹)
22		inss2 4246	. . . . . . . 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 773	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
25		filelss 23876	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
26	25	3ad2antl1 1184	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
27	26	ad2ant2r 747	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝐴)
28	24, 27	ssind 4249	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ (𝑥 ∩ 𝐴))
29		filss 23877	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ (𝑦 ∈ 𝐹 ∧ (𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝑦 ⊆ (𝑥 ∩ 𝐴))) → (𝑥 ∩ 𝐴) ∈ 𝐹)
30	20, 21, 23, 28, 29	syl13anc 1371	. . . . . 6 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
31	19, 30	rexlimddv 3159	. . . . 5 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
32	31	fmpttd 7135	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)):(𝑋filGen𝐹)⟶𝐹)
33	32	frnd 6745	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)) ⊆ 𝐹)
34	16, 33	eqsstrd 4034	. 2 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) ⊆ 𝐹)
35		filelss 23876	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
36	35	3ad2antl1 1184	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
37		dfss2 3981	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝑥 ∩ 𝐴) = 𝑥)
38	36, 37	sylib 218	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) = 𝑥)
39	12	adantr 480	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
40	14	adantr 480	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝐴 ∈ 𝐹)
41		ssfg 23896	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
42	10, 41	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ (𝑋filGen𝐹))
43	42	sselda 3995	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝑋filGen𝐹))
44		elrestr 17475	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
45	39, 40, 43, 44	syl3anc 1370	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
46	38, 45	eqeltrrd 2840	. 2 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ ((𝑋filGen𝐹) ↾t 𝐴))
47	34, 46	eqelssd 4017	1 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃wrex 3068   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605   ↦ cmpt 5231  ran crn 5690  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  fBascfbas 21370  filGencfg 21371  Filcfil 23869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17469  df-fbas 21379  df-fg 21380  df-fil 23870

This theorem is referenced by:  cmetss  25364  minveclem4a  25478