Theorem lanpropd 50105

Description: If the categories have the same set of objects, morphisms, and compositions, then they have the same left Kan extensions. (Contributed by Zhi Wang, 21-Nov-2025.)

Hypotheses

Ref	Expression
lanpropd.1	⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
lanpropd.2	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
lanpropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
lanpropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
lanpropd.5	⊢ (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))
lanpropd.6	⊢ (𝜑 → (compf‘𝐸) = (compf‘𝐹))
lanpropd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
lanpropd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
lanpropd.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
lanpropd.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
lanpropd.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
lanpropd.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)

Assertion

Ref	Expression
lanpropd	⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (⟨𝐵, 𝐷⟩ Lan 𝐹))

Proof of Theorem lanpropd

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lanpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2		lanpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
3		lanpropd.3	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4		lanpropd.4	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
5		lanpropd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		lanpropd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		lanpropd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
8		lanpropd.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 17860	. . 3 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10		lanpropd.5	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐸) = (Homf ‘𝐹))
11		lanpropd.6	. . . . 5 ⊢ (𝜑 → (compf‘𝐸) = (compf‘𝐹))
12		lanpropd.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
13		lanpropd.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
14	1, 2, 10, 11, 5, 6, 12, 13	funcpropd 17860	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐸) = (𝐵 Func 𝐹))
15	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐸) = (𝐵 Func 𝐹))
16	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐶) = (Homf ‘𝐷))
17	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐶) = (compf‘𝐷))
18	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐸) = (Homf ‘𝐹))
19	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐸) = (compf‘𝐹))
20		funcrcl 17821	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
21	20	ad2antrl 734	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
22	21	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐶 ∈ Cat)
23	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ 𝑉)
24	16, 17, 22, 23	catpropd 17666	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
25	22, 24	mpbid 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ Cat)
26		funcrcl 17821	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 Func 𝐸) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
27	26	ad2antll 735	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
28	27	simprd 496	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐸 ∈ Cat)
29	13	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ 𝑉)
30	18, 19, 28, 29	catpropd 17666	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐸 ∈ Cat ↔ 𝐹 ∈ Cat))
31	28, 30	mpbid 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ Cat)
32	16, 17, 18, 19, 22, 25, 28, 31	fucpropd 17938	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 FuncCat 𝐸) = (𝐷 FuncCat 𝐹))
33	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐴) = (Homf ‘𝐵))
34	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐴) = (compf‘𝐵))
35	21	simpld 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐴 ∈ Cat)
36	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ 𝑉)
37	33, 34, 35, 36	catpropd 17666	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ↔ 𝐵 ∈ Cat))
38	35, 37	mpbid 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ Cat)
39	33, 34, 18, 19, 35, 38, 28, 31	fucpropd 17938	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 FuncCat 𝐸) = (𝐵 FuncCat 𝐹))
40	32, 39	oveq12d 7374	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → ((𝐶 FuncCat 𝐸) UP (𝐴 FuncCat 𝐸)) = ((𝐷 FuncCat 𝐹) UP (𝐵 FuncCat 𝐹)))
41		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝑓 ∈ (𝐴 Func 𝐶))
42	16, 17, 18, 19, 22, 25, 28, 31, 41	prcofpropd 49869	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (⟨𝐶, 𝐸⟩ −∘F 𝑓) = (⟨𝐷, 𝐹⟩ −∘F 𝑓))
43		eqidd 2740	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝑥 = 𝑥)
44	40, 42, 43	oveq123d 7377	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → ((⟨𝐶, 𝐸⟩ −∘F 𝑓)((𝐶 FuncCat 𝐸) UP (𝐴 FuncCat 𝐸))𝑥) = ((⟨𝐷, 𝐹⟩ −∘F 𝑓)((𝐷 FuncCat 𝐹) UP (𝐵 FuncCat 𝐹))𝑥))
45	9, 15, 44	mpoeq123dva 7430	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 Func 𝐶), 𝑥 ∈ (𝐴 Func 𝐸) ↦ ((⟨𝐶, 𝐸⟩ −∘F 𝑓)((𝐶 FuncCat 𝐸) UP (𝐴 FuncCat 𝐸))𝑥)) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑥 ∈ (𝐵 Func 𝐹) ↦ ((⟨𝐷, 𝐹⟩ −∘F 𝑓)((𝐷 FuncCat 𝐹) UP (𝐵 FuncCat 𝐹))𝑥)))
46		eqid 2739	. . 3 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
47		eqid 2739	. . 3 ⊢ (𝐴 FuncCat 𝐸) = (𝐴 FuncCat 𝐸)
48	46, 47, 5, 7, 12	lanfval 50103	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (𝑓 ∈ (𝐴 Func 𝐶), 𝑥 ∈ (𝐴 Func 𝐸) ↦ ((⟨𝐶, 𝐸⟩ −∘F 𝑓)((𝐶 FuncCat 𝐸) UP (𝐴 FuncCat 𝐸))𝑥)))
49		eqid 2739	. . 3 ⊢ (𝐷 FuncCat 𝐹) = (𝐷 FuncCat 𝐹)
50		eqid 2739	. . 3 ⊢ (𝐵 FuncCat 𝐹) = (𝐵 FuncCat 𝐹)
51	49, 50, 6, 8, 13	lanfval 50103	. 2 ⊢ (𝜑 → (⟨𝐵, 𝐷⟩ Lan 𝐹) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑥 ∈ (𝐵 Func 𝐹) ↦ ((⟨𝐷, 𝐹⟩ −∘F 𝑓)((𝐷 FuncCat 𝐹) UP (𝐵 FuncCat 𝐹))𝑥)))
52	45, 48, 51	3eqtr4d 2784	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (⟨𝐵, 𝐷⟩ Lan 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4561  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  Catccat 17621  Hom_f chomf 17623  comp_fccomf 17624   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-tp 4560  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-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-ixp 8836  df-cat 17625  df-cid 17626  df-homf 17627  df-comf 17628  df-func 17816  df-nat 17904  df-fuc 17905  df-prcof 49864  df-lan 50097

This theorem is referenced by: (None)