Theorem rellan 50421

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 5785	. . 3 ⊢ Rel ∅
2		releq 5763	. . 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 4307	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
5		relup 49981	. . . . 5 ⊢ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋)
6		ne0i 4294	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅)
7		oveq 7416	. . . . . . . . . . . 12 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7447	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = ∅)
10	9	necon3i 2990	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → (𝑃 Lan 𝐸) ≠ ∅)
11		n0 4307	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸))
12		df-lan 50405	. . . . . . . . . . . . . 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 7652	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 8021	. . . . . . . . . . . . 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 1960	. . . . . . . . . . 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 7425	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝑃 Lan 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸))
20	19	oveqd 7427	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
21		eqid 2763	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2763	. . . . . . . 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 2865	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
25		lanrcl 50419	. . . . . . . . . 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 2764	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))
30	21, 22, 27, 28, 29	lanval 50417	. . . . . . 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 2798	. . . . . 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 5765	. . . . 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 1960	. . 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 3041	1 ⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ⦋csb 3853  ∅c0 4286  ⟨cop 4595   × cxp 5659  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981   Func cfunc 17906   FuncCat cfuc 17997   UP cup 49971   −∘_F cprcof 50171   Lan clan 50403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-func 17910  df-up 49972  df-lan 50405

This theorem is referenced by: (None)