Theorem rellan 50113

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 5742	. . 3 ⊢ Rel ∅
2		releq 5720	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃 Lan 𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 259	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
4		n0 4281	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
5		relup 49673	. . . . 5 ⊢ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋)
6		ne0i 4269	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅)
7		oveq 7362	. . . . . . . . . . . 12 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7393	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2790	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = ∅)
10	9	necon3i 2966	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → (𝑃 Lan 𝐸) ≠ ∅)
11		n0 4281	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸))
12		df-lan 50097	. . . . . . . . . . . . . 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 7598	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 7970	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
15	13, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
16	15	exlimiv 1937	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 218	. . . . . . . . . 10 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
18	6, 10, 17	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	18	oveq1d 7371	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝑃 Lan 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸))
20	19	oveqd 7373	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
21		eqid 2739	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2739	. . . . . . . 8 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
24	23, 20	eleqtrd 2841	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
25		lanrcl 50111	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
27	26	simpld 495	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 496	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		eqidd 2740	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))
30	21, 22, 27, 28, 29	lanval 50109	. . . . . . 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 2774	. . . . . 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 5722	. . . . 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 259	. . . 4 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
34	33	exlimiv 1937	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
35	4, 34	sylbi 218	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
36	3, 35	pm2.61ine 3017	1 ⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)

Syntax hints:   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  Vcvv 3431  ⦋csb 3831  ∅c0 4261  ⟨cop 4561   × cxp 5616  Rel wrel 5623  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930   Func cfunc 17812   FuncCat cfuc 17903   UP cup 49663   −∘_F cprcof 49863   Lan clan 50095

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-func 17816  df-up 49664  df-lan 50097

This theorem is referenced by: (None)