Theorem rellan 50098

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 5755	. . 3 ⊢ Rel ∅
2		releq 5733	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃 Lan 𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 258	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
4		n0 4293	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
5		relup 49658	. . . . 5 ⊢ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋)
6		ne0i 4281	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅)
7		oveq 7373	. . . . . . . . . . . 12 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7404	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2787	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = ∅)
10	9	necon3i 2964	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → (𝑃 Lan 𝐸) ≠ ∅)
11		n0 4293	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸))
12		df-lan 50082	. . . . . . . . . . . . . 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 7609	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 7981	. . . . . . . . . . . . 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 1932	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 217	. . . . . . . . . 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 7382	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝑃 Lan 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸))
20	19	oveqd 7384	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
21		eqid 2736	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2736	. . . . . . . 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 2838	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
25		lanrcl 50096	. . . . . . . . . 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 494	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 495	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		eqidd 2737	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))
30	21, 22, 27, 28, 29	lanval 50094	. . . . . . 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 2771	. . . . . 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 5735	. . . . 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 258	. . . 4 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
34	33	exlimiv 1932	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
35	4, 34	sylbi 217	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
36	3, 35	pm2.61ine 3015	1 ⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429  ⦋csb 3837  ∅c0 4273  ⟨cop 4573   × cxp 5629  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941   Func cfunc 17821   FuncCat cfuc 17912   UP cup 49648   −∘_F cprcof 49848   Lan clan 50080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-func 17825  df-up 49649  df-lan 50082

This theorem is referenced by: (None)