Theorem rellan 49359

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

Assertion

Ref	Expression
rellan	⊢ Rel (𝐹(𝑃Lan𝐸)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 5776	. . 3 ⊢ Rel ∅
2		releq 5753	. . 3 ⊢ ((𝐹(𝑃Lan𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃Lan𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 258	. 2 ⊢ ((𝐹(𝑃Lan𝐸)𝑋) = ∅ → Rel (𝐹(𝑃Lan𝐸)𝑋))
4		n0 4326	. . 3 ⊢ ((𝐹(𝑃Lan𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋))
5		relup 48983	. . . . 5 ⊢ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸)UP((1st ‘𝑃) FuncCat 𝐸))𝑋)
6		ne0i 4314	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (𝐹(𝑃Lan𝐸)𝑋) ≠ ∅)
7		oveq 7406	. . . . . . . . . . . 12 ⊢ ((𝑃Lan𝐸) = ∅ → (𝐹(𝑃Lan𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7437	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2785	. . . . . . . . . . 11 ⊢ ((𝑃Lan𝐸) = ∅ → (𝐹(𝑃Lan𝐸)𝑋) = ∅)
10	9	necon3i 2963	. . . . . . . . . 10 ⊢ ((𝐹(𝑃Lan𝐸)𝑋) ≠ ∅ → (𝑃Lan𝐸) ≠ ∅)
11		n0 4326	. . . . . . . . . . 11 ⊢ ((𝑃Lan𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃Lan𝐸))
12		df-lan 49345	. . . . . . . . . . . . . 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 7644	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃Lan𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 8022	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (V × V) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
15	13, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑃Lan𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
16	15	exlimiv 1929	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃Lan𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 217	. . . . . . . . . 10 ⊢ ((𝑃Lan𝐸) ≠ ∅ → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
18	6, 10, 17	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
19	18	oveq1d 7415	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (𝑃Lan𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Lan𝐸))
20	19	oveqd 7417	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (𝐹(𝑃Lan𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Lan𝐸)𝑋))
21		eqid 2734	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2734	. . . . . . . 8 ⊢ ((1st ‘𝑃) FuncCat 𝐸) = ((1st ‘𝑃) FuncCat 𝐸)
23		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → 𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋))
24	23, 20	eleqtrd 2835	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Lan𝐸)𝑋))
25		lanrcl 49357	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Lan𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)) ∧ 𝑋 ∈ ((1st ‘𝑃) Func 𝐸)))
27	26	simpld 494	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 495	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		eqidd 2735	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))
30	21, 22, 27, 28, 29	lanval 49355	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩Lan𝐸)𝑋) = ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸)UP((1st ‘𝑃) FuncCat 𝐸))𝑋))
31	20, 30	eqtrd 2769	. . . . . 6 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (𝐹(𝑃Lan𝐸)𝑋) = ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸)UP((1st ‘𝑃) FuncCat 𝐸))𝑋))
32	31	releqd 5755	. . . . 5 ⊢ (𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → (Rel (𝐹(𝑃Lan𝐸)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(𝑃Lan𝐸)𝑋) → Rel (𝐹(𝑃Lan𝐸)𝑋))
34	33	exlimiv 1929	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃Lan𝐸)𝑋) → Rel (𝐹(𝑃Lan𝐸)𝑋))
35	4, 34	sylbi 217	. 2 ⊢ ((𝐹(𝑃Lan𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃Lan𝐸)𝑋))
36	3, 35	pm2.61ine 3014	1 ⊢ Rel (𝐹(𝑃Lan𝐸)𝑋)

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2931  Vcvv 3457  ⦋csb 3872  ∅c0 4306  ⟨cop 4605   × cxp 5650  Rel wrel 5657  ‘cfv 6528  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7981  2^nd c2nd 7982   Func cfunc 17854   FuncCat cfuc 17945  UPcup 48974   −∘_F cprcof 49147  Lanclan 49343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7983  df-2nd 7984  df-func 17858  df-up 48975  df-lan 49345

This theorem is referenced by: (None)