Theorem ranpropd 51130

Description: If the categories have the same set of objects, morphisms, and compositions, then they have the same right 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
ranpropd	⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (⟨𝐵, 𝐷⟩ Ran 𝐹))

Proof of Theorem ranpropd

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 18127	. . 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 18127	. . . 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	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐶) = (Homf ‘𝐷))
17	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐶) = (compf‘𝐷))
18	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐸) = (Homf ‘𝐹))
19	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐸) = (compf‘𝐹))
20		funcrcl 18088	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
21	20	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
22	21	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐶 ∈ Cat)
23	8	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ 𝑉)
24	16, 17, 22, 23	catpropd 17933	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
25	22, 24	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ Cat)
26		funcrcl 18088	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 Func 𝐸) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
27	26	ad2antll 742	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
28	27	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐸 ∈ Cat)
29	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ 𝑉)
30	18, 19, 28, 29	catpropd 17933	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐸 ∈ Cat ↔ 𝐹 ∈ Cat))
31	28, 30	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ Cat)
32	16, 17, 18, 19, 22, 25, 28, 31	fucpropd 18205	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 FuncCat 𝐸) = (𝐷 FuncCat 𝐹))
33	32	fveq2d 6920	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐹)))
34	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐴) = (Homf ‘𝐵))
35	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐴) = (compf‘𝐵))
36	21	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐴 ∈ Cat)
37	6	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ 𝑉)
38	34, 35, 36, 37	catpropd 17933	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ↔ 𝐵 ∈ Cat))
39	36, 38	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ Cat)
40	34, 35, 18, 19, 36, 39, 28, 31	fucpropd 18205	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 FuncCat 𝐸) = (𝐵 FuncCat 𝐹))
41	40	fveq2d 6920	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (oppCat‘(𝐴 FuncCat 𝐸)) = (oppCat‘(𝐵 FuncCat 𝐹)))
42	33, 41	oveq12d 7469	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → ((oppCat‘(𝐶 FuncCat 𝐸)) UP (oppCat‘(𝐴 FuncCat 𝐸))) = ((oppCat‘(𝐷 FuncCat 𝐹)) UP (oppCat‘(𝐵 FuncCat 𝐹))))
43		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝑓 ∈ (𝐴 Func 𝐶))
44	16, 17, 18, 19, 22, 25, 28, 31, 43	prcofpropd 50845	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (⟨𝐶, 𝐸⟩ −∘F 𝑓) = (⟨𝐷, 𝐹⟩ −∘F 𝑓))
45	44	fveq2d 6920	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → ( oppFunc ‘(⟨𝐶, 𝐸⟩ −∘F 𝑓)) = ( oppFunc ‘(⟨𝐷, 𝐹⟩ −∘F 𝑓)))
46		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝑥 = 𝑥)
47	42, 45, 46	oveq123d 7472	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (( oppFunc ‘(⟨𝐶, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐶 FuncCat 𝐸)) UP (oppCat‘(𝐴 FuncCat 𝐸)))𝑥) = (( oppFunc ‘(⟨𝐷, 𝐹⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐹)) UP (oppCat‘(𝐵 FuncCat 𝐹)))𝑥))
48	9, 15, 47	mpoeq123dva 7525	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐴 Func 𝐶), 𝑥 ∈ (𝐴 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐶, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐶 FuncCat 𝐸)) UP (oppCat‘(𝐴 FuncCat 𝐸)))𝑥)) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑥 ∈ (𝐵 Func 𝐹) ↦ (( oppFunc ‘(⟨𝐷, 𝐹⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐹)) UP (oppCat‘(𝐵 FuncCat 𝐹)))𝑥)))
49		eqid 2763	. . 3 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
50		eqid 2763	. . 3 ⊢ (𝐴 FuncCat 𝐸) = (𝐴 FuncCat 𝐸)
51		eqid 2763	. . 3 ⊢ (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐶 FuncCat 𝐸))
52		eqid 2763	. . 3 ⊢ (oppCat‘(𝐴 FuncCat 𝐸)) = (oppCat‘(𝐴 FuncCat 𝐸))
53	49, 50, 5, 7, 12, 51, 52	ranfval 51128	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (𝑓 ∈ (𝐴 Func 𝐶), 𝑥 ∈ (𝐴 Func 𝐸) ↦ (( oppFunc ‘(⟨𝐶, 𝐸⟩ −∘F 𝑓))((oppCat‘(𝐶 FuncCat 𝐸)) UP (oppCat‘(𝐴 FuncCat 𝐸)))𝑥)))
54		eqid 2763	. . 3 ⊢ (𝐷 FuncCat 𝐹) = (𝐷 FuncCat 𝐹)
55		eqid 2763	. . 3 ⊢ (𝐵 FuncCat 𝐹) = (𝐵 FuncCat 𝐹)
56		eqid 2763	. . 3 ⊢ (oppCat‘(𝐷 FuncCat 𝐹)) = (oppCat‘(𝐷 FuncCat 𝐹))
57		eqid 2763	. . 3 ⊢ (oppCat‘(𝐵 FuncCat 𝐹)) = (oppCat‘(𝐵 FuncCat 𝐹))
58	54, 55, 6, 8, 13, 56, 57	ranfval 51128	. 2 ⊢ (𝜑 → (⟨𝐵, 𝐷⟩ Ran 𝐹) = (𝑓 ∈ (𝐵 Func 𝐷), 𝑥 ∈ (𝐵 Func 𝐹) ↦ (( oppFunc ‘(⟨𝐷, 𝐹⟩ −∘F 𝑓))((oppCat‘(𝐷 FuncCat 𝐹)) UP (oppCat‘(𝐵 FuncCat 𝐹)))𝑥)))
59	48, 53, 58	3eqtr4d 2808	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (⟨𝐵, 𝐷⟩ Ran 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Catccat 17888  Hom_f chomf 17890  comp_fccomf 17891  oppCatcoppc 17935   Func cfunc 18079   FuncCat cfuc 18170   oppFunc coppf 50569   UP cup 50620   −∘_F cprcof 50839   Ran cran 51120

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-ixp 8964  df-cat 17892  df-cid 17893  df-homf 17894  df-comf 17895  df-func 18083  df-nat 18171  df-fuc 18172  df-prcof 50840  df-ran 51122

This theorem is used by: (None)