Theorem rellan 50244

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 5771	. . 3 ⊢ Rel ∅
2		releq 5749	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → (Rel (𝐹(𝑃 Lan 𝐸)𝑋) ↔ Rel ∅))
3	1, 2	mpbiri 260	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) = ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
4		n0 4305	. . 3 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋))
5		relup 49804	. . . . 5 ⊢ Rel ((⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹)(((2nd ‘𝑃) FuncCat 𝐸) UP ((1st ‘𝑃) FuncCat 𝐸))𝑋)
6		ne0i 4293	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅)
7		oveq 7402	. . . . . . . . . . . 12 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹∅𝑋))
8		0ov 7433	. . . . . . . . . . . 12 ⊢ (𝐹∅𝑋) = ∅
9	7, 8	eqtrdi 2813	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) = ∅ → (𝐹(𝑃 Lan 𝐸)𝑋) = ∅)
10	9	necon3i 2989	. . . . . . . . . 10 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → (𝑃 Lan 𝐸) ≠ ∅)
11		n0 4305	. . . . . . . . . . 11 ⊢ ((𝑃 Lan 𝐸) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸))
12		df-lan 50228	. . . . . . . . . . . . . 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 7638	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 ∈ (V × V))
14		1st2nd2 8009	. . . . . . . . . . . . 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 1950	. . . . . . . . . . 11 ⊢ (∃𝑥 𝑥 ∈ (𝑃 Lan 𝐸) → 𝑃 = ⟨(1st ‘𝑃), (2nd ‘𝑃)⟩)
17	11, 16	sylbi 219	. . . . . . . . . 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 7411	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝑃 Lan 𝐸) = (⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸))
20	19	oveqd 7413	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (𝐹(𝑃 Lan 𝐸)𝑋) = (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
21		eqid 2762	. . . . . . . 8 ⊢ ((2nd ‘𝑃) FuncCat 𝐸) = ((2nd ‘𝑃) FuncCat 𝐸)
22		eqid 2762	. . . . . . . 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 2864	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑥 ∈ (𝐹(⟨(1st ‘𝑃), (2nd ‘𝑃)⟩ Lan 𝐸)𝑋))
25		lanrcl 50242	. . . . . . . . . 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 498	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝐹 ∈ ((1st ‘𝑃) Func (2nd ‘𝑃)))
28	26	simprd 499	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → 𝑋 ∈ ((1st ‘𝑃) Func 𝐸))
29		eqidd 2763	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹) = (⟨(2nd ‘𝑃), 𝐸⟩ −∘F 𝐹))
30	21, 22, 27, 28, 29	lanval 50240	. . . . . . 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 2797	. . . . . 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 5751	. . . . 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 260	. . . 4 ⊢ (𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
34	33	exlimiv 1950	. . 3 ⊢ (∃𝑥 𝑥 ∈ (𝐹(𝑃 Lan 𝐸)𝑋) → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
35	4, 34	sylbi 219	. 2 ⊢ ((𝐹(𝑃 Lan 𝐸)𝑋) ≠ ∅ → Rel (𝐹(𝑃 Lan 𝐸)𝑋))
36	3, 35	pm2.61ine 3040	1 ⊢ Rel (𝐹(𝑃 Lan 𝐸)𝑋)

Syntax hints:   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454  ⦋csb 3852  ∅c0 4285  ⟨cop 4588   × cxp 5645  Rel wrel 5652  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  1^st c1st 7968  2^nd c2nd 7969   Func cfunc 17887   FuncCat cfuc 17978   UP cup 49794   −∘_F cprcof 49994   Lan clan 50226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-func 17891  df-up 49795  df-lan 50228

This theorem is referenced by: (None)