Theorem lanpropd 50542

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 17992	. . 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 17992	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐸) = (𝐵 Func 𝐹))
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐸) = (𝐵 Func 𝐹))
16	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐶) = (Homf ‘𝐷))
17	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐶) = (compf‘𝐷))
18	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐸) = (Homf ‘𝐹))
19	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐸) = (compf‘𝐹))
20		funcrcl 17953	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
21	20	ad2antrl 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
22	21	simprd 501	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐶 ∈ Cat)
23	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ 𝑉)
24	16, 17, 22, 23	catpropd 17798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
25	22, 24	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ Cat)
26		funcrcl 17953	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 Func 𝐸) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
27	26	ad2antll 742	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
28	27	simprd 501	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐸 ∈ Cat)
29	13	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ 𝑉)
30	18, 19, 28, 29	catpropd 17798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐸 ∈ Cat ↔ 𝐹 ∈ Cat))
31	28, 30	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ Cat)
32	16, 17, 18, 19, 22, 25, 28, 31	fucpropd 18070	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 FuncCat 𝐸) = (𝐷 FuncCat 𝐹))
33	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐴) = (Homf ‘𝐵))
34	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐴) = (compf‘𝐵))
35	21	simpld 500	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐴 ∈ Cat)
36	6	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ 𝑉)
37	33, 34, 35, 36	catpropd 17798	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ↔ 𝐵 ∈ Cat))
38	35, 37	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ Cat)
39	33, 34, 18, 19, 35, 38, 28, 31	fucpropd 18070	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 FuncCat 𝐸) = (𝐵 FuncCat 𝐹))
40	32, 39	oveq12d 7432	. . . 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 783	. . . . 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 50306	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (⟨𝐶, 𝐸⟩ −∘F 𝑓) = (⟨𝐷, 𝐹⟩ −∘F 𝑓))
43		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝑥 = 𝑥)
44	40, 42, 43	oveq123d 7435	. . 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 7488	. 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 2760	. . 3 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
47		eqid 2760	. . 3 ⊢ (𝐴 FuncCat 𝐸) = (𝐴 FuncCat 𝐸)
48	46, 47, 5, 7, 12	lanfval 50540	. 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 2760	. . 3 ⊢ (𝐷 FuncCat 𝐹) = (𝐷 FuncCat 𝐹)
50		eqid 2760	. . 3 ⊢ (𝐵 FuncCat 𝐹) = (𝐵 FuncCat 𝐹)
51	49, 50, 6, 8, 13	lanfval 50540	. 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 2805	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Lan 𝐸) = (⟨𝐵, 𝐷⟩ Lan 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  Catccat 17753  Hom_f chomf 17755  comp_fccomf 17756   Func cfunc 17944   FuncCat cfuc 18035   UP cup 50100   −∘_F cprcof 50300   Lan clan 50532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8829  df-ixp 8906  df-cat 17757  df-cid 17758  df-homf 17759  df-comf 17760  df-func 17948  df-nat 18036  df-fuc 18037  df-prcof 50301  df-lan 50534

This theorem is used by: (None)