Theorem trfg 24210

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 24209 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 24167	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ∈ (fBas‘𝐴))
2	1	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝐴))
3		filsspw 24170	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐹 ⊆ 𝒫 𝐴)
4	3	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝐴)
5		simp2 1155	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ⊆ 𝑋)
6	5	sspwd 4570	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝒫 𝐴 ⊆ 𝒫 𝑋)
7	4, 6	sstrd 3941	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ 𝒫 𝑋)
8		simp3 1156	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
9		fbasweak 24184	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝐴) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
10	2, 7, 8, 9	syl3anc 1398	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑋))
11		fgcl 24197	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
12	10, 11	syl 18	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
13		filtop 24174	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝐴) → 𝐴 ∈ 𝐹)
14	13	3ad2ant1 1151	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐴 ∈ 𝐹)
15		restval 17597	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → ((𝑋filGen𝐹) ↾t 𝐴) = ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)))
16	12, 14, 15	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)))
17		elfg 24190	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐹) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)))
18	10, 17	syl 18	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝑋filGen𝐹) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)))
19	18	simplbda 505	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥)
20		simpll1 1231	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝐴))
21		simprl 783	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐹)
22		inss2 4183	. . . . . . . 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 785	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
25		filelss 24171	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
26	25	3ad2antl1 1204	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝐴)
27	26	ad2ant2r 760	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝐴)
28	24, 27	ssind 4186	. . . . . . 7 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ (𝑥 ∩ 𝐴))
29		filss 24172	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ (𝑦 ∈ 𝐹 ∧ (𝑥 ∩ 𝐴) ⊆ 𝐴 ∧ 𝑦 ⊆ (𝑥 ∩ 𝐴))) → (𝑥 ∩ 𝐴) ∈ 𝐹)
30	20, 21, 23, 28, 29	syl13anc 1399	. . . . . 6 ⊢ ((((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) ∧ (𝑦 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑥)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
31	19, 30	rexlimddv 3170	. . . . 5 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ 𝐹)
32	31	fmpttd 7115	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)):(𝑋filGen𝐹)⟶𝐹)
33	32	frnd 6718	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ran (𝑥 ∈ (𝑋filGen𝐹) ↦ (𝑥 ∩ 𝐴)) ⊆ 𝐹)
34	16, 33	eqsstrd 3965	. 2 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) ⊆ 𝐹)
35		filelss 24171	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
36	35	3ad2antl1 1204	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ⊆ 𝐴)
37		dfss2 3917	. . . 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 486	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
40	14	adantr 486	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝐴 ∈ 𝐹)
41		ssfg 24191	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
42	10, 41	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → 𝐹 ⊆ (𝑋filGen𝐹))
43	42	sselda 3931	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ (𝑋filGen𝐹))
44		elrestr 17599	. . . 4 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝑥 ∈ (𝑋filGen𝐹)) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
45	39, 40, 43, 44	syl3anc 1398	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → (𝑥 ∩ 𝐴) ∈ ((𝑋filGen𝐹) ↾t 𝐴))
46	38, 45	eqeltrrd 2862	. 2 ⊢ (((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝐹) → 𝑥 ∈ ((𝑋filGen𝐹) ↾t 𝐴))
47	34, 46	eqelssd 3952	1 ⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557   ↦ cmpt 5186  ran crn 5652  ‘cfv 6538  (class class class)co 7420   ↾_t crest 17591  fBascfbas 21666  filGencfg 21667  Filcfil 24164

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rest 17593  df-fbas 21675  df-fg 21676  df-fil 24165

This theorem is used by:  cmetss  25637  minveclem4a  25751