Theorem rellan 50286

Description: The set of left Kan extensions is a relation. (Contributed by Zhi Wang, 3-Nov-2025.)

Assertion

Ref	Expression
rellan	⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)

Proof of Theorem rellan

Dummy variables 𝑓 𝑥 𝑐 𝑑 𝑒 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 5786	. . 3 ⊢ Rel ∅
2		releq 5764	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃 Lan 𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 261	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
4		n0 4315	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
5		relup 49846	. . . . 5 ⊢ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋)
6		ne0i 4302	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅)
7		oveq 7417	. . . . . . . . . . . 12 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7448	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2820	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = ∅)
10	9	necon3i 2996	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → (𝑃 Lan 𝐸) ≠ ∅)
11		n0 4315	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸))
12		df-lan 50270	. . . . . . . . . . . . . 14 ⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)))
13	12	elmpocl1 7653	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 8025	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
15	13, 14	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
16	15	exlimiv 1957	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 220	. . . . . . . . . 10 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
18	6, 10, 17	3syl 19	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	18	oveq1d 7426	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝑃 Lan 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸))
20	19	oveqd 7428	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
21		eqid 2769	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2769	. . . . . . . 8 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		id 23	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
24	23, 20	eleqtrd 2871	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
25		lanrcl 50284	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
26	24, 25	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
27	26	simpld 499	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 500	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		eqidd 2770	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))
30	21, 22, 27, 28, 29	lanval 50282	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋) = ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋))
31	20, 30	eqtrd 2804	. . . . . 6 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) = ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋))
32	31	releqd 5766	. . . . 5 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (Rel (𝐹(𝑃 Lan 𝐸)𝑋) ↔ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋)))
33	5, 32	mpbiri 261	. . . 4 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
34	33	exlimiv 1957	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
35	4, 34	sylbi 220	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
36	3, 35	pm2.61ine 3047	1 ⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ⦋csb 3861  ∅c0 4294  ⟨cop 4600   × cxp 5660  Rel wrel 5667  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985   Func cfunc 17911   FuncCat cfuc 18002   UP cup 49836   −∘_F cprcof 50036   Lan clan 50268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-func 17915  df-up 49837  df-lan 50270

This theorem is referenced by: (None)