Theorem trfg 22654

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 22653 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 22611	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ∈ (fBas‘𝐴))
2	1	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝐴))
3		filsspw 22614	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
4	3	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝐴)
5		simp2 1138	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ⊆ 𝑋)
6	5	sspwd 4513	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
7	4, 6	sstrd 3897	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝑋)
8		simp3 1139	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
9		fbasweak 22628	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
10	2, 7, 8, 9	syl3anc 1372	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
11		fgcl 22641	. . . . 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 22618	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐴 ∈ 𝐹)
14	13	3ad2ant1 1134	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ∈ 𝐹)
15		restval 16815	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝑋filGen𝐹) ↾t 𝐴) = ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)))
16	12, 14, 15	syl2anc 587	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)))
17		elfg 22634	. . . . . . . 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 503	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)
20		simpll1 1213	. . . . . . 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 4130	. . . . . . . 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 22615	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
26	25	3ad2antl1 1186	. . . . . . . . 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 4133	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ (𝑥 ∩ 𝐴))
29		filss 22616	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ (𝑦 ∈ 𝐹 ∧ (𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝑦 ⊆ (𝑥 ∩ 𝐴))) → (𝑥 ∩ 𝐴) ∈ 𝐹)
30	20, 21, 23, 28, 29	syl13anc 1373	. . . . . 6 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
31	19, 30	rexlimddv 3202	. . . . 5 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
32	31	fmpttd 6901	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)):(𝑋filGen𝐹)⟶𝐹)
33	32	frnd 6522	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)) ⊆ 𝐹)
34	16, 33	eqsstrd 3925	. 2 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) ⊆ 𝐹)
35		filelss 22615	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
36	35	3ad2antl1 1186	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
37		df-ss 3870	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝑥 ∩ 𝐴) = 𝑥)
38	36, 37	sylib 221	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) = 𝑥)
39	12	adantr 484	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
40	14	adantr 484	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝐴 ∈ 𝐹)
41		ssfg 22635	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
42	10, 41	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ (𝑋filGen𝐹))
43	42	sselda 3887	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝑋filGen𝐹))
44		elrestr 16817	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
45	39, 40, 43, 44	syl3anc 1372	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
46	38, 45	eqeltrrd 2835	. 2 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ ((𝑋filGen𝐹) ↾t 𝐴))
47	34, 46	eqelssd 3908	1 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∃wrex 3055   ∩ cin 3852   ⊆ wss 3853  𝒫 cpw 4498   ↦ cmpt 5120  ran crn 5536  ‘cfv 6349  (class class class)co 7182   ↾_t crest 16809  fBascfbas 20217  filGencfg 20218  Filcfil 22608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-rest 16811  df-fbas 20226  df-fg 20227  df-fil 22609

This theorem is referenced by:  cmetss  24080  minveclem4a  24194