Theorem ranpropd 49598

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 17844	. . 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 17844	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐸) = (𝐵 Func 𝐹))
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐸) = (𝐵 Func 𝐹))
16	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐶) = (Homf ‘𝐷))
17	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐶) = (compf‘𝐷))
18	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐸) = (Homf ‘𝐹))
19	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐸) = (compf‘𝐹))
20		funcrcl 17805	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
21	20	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
22	21	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐶 ∈ Cat)
23	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ 𝑉)
24	16, 17, 22, 23	catpropd 17650	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
25	22, 24	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐷 ∈ Cat)
26		funcrcl 17805	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 Func 𝐸) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
27	26	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ∧ 𝐸 ∈ Cat))
28	27	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐸 ∈ Cat)
29	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ 𝑉)
30	18, 19, 28, 29	catpropd 17650	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐸 ∈ Cat ↔ 𝐹 ∈ Cat))
31	28, 30	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐹 ∈ Cat)
32	16, 17, 18, 19, 22, 25, 28, 31	fucpropd 17922	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐶 FuncCat 𝐸) = (𝐷 FuncCat 𝐹))
33	32	fveq2d 6844	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐹)))
34	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (Homf ‘𝐴) = (Homf ‘𝐵))
35	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (compf‘𝐴) = (compf‘𝐵))
36	21	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐴 ∈ Cat)
37	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ 𝑉)
38	34, 35, 36, 37	catpropd 17650	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 ∈ Cat ↔ 𝐵 ∈ Cat))
39	36, 38	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝐵 ∈ Cat)
40	34, 35, 18, 19, 36, 39, 28, 31	fucpropd 17922	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (𝐴 FuncCat 𝐸) = (𝐵 FuncCat 𝐹))
41	40	fveq2d 6844	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (oppCat‘(𝐴 FuncCat 𝐸)) = (oppCat‘(𝐵 FuncCat 𝐹)))
42	33, 41	oveq12d 7387	. . . 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 770	. . . . . 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 49361	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → (⟨𝐶, 𝐸⟩ −∘F 𝑓) = (⟨𝐷, 𝐹⟩ −∘F 𝑓))
45	44	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → ( oppFunc ‘(⟨𝐶, 𝐸⟩ −∘F 𝑓)) = ( oppFunc ‘(⟨𝐷, 𝐹⟩ −∘F 𝑓)))
46		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝐴 Func 𝐶) ∧ 𝑥 ∈ (𝐴 Func 𝐸))) → 𝑥 = 𝑥)
47	42, 45, 46	oveq123d 7390	. . 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 7443	. 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 2729	. . 3 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
50		eqid 2729	. . 3 ⊢ (𝐴 FuncCat 𝐸) = (𝐴 FuncCat 𝐸)
51		eqid 2729	. . 3 ⊢ (oppCat‘(𝐶 FuncCat 𝐸)) = (oppCat‘(𝐶 FuncCat 𝐸))
52		eqid 2729	. . 3 ⊢ (oppCat‘(𝐴 FuncCat 𝐸)) = (oppCat‘(𝐴 FuncCat 𝐸))
53	49, 50, 5, 7, 12, 51, 52	ranfval 49596	. 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 2729	. . 3 ⊢ (𝐷 FuncCat 𝐹) = (𝐷 FuncCat 𝐹)
55		eqid 2729	. . 3 ⊢ (𝐵 FuncCat 𝐹) = (𝐵 FuncCat 𝐹)
56		eqid 2729	. . 3 ⊢ (oppCat‘(𝐷 FuncCat 𝐹)) = (oppCat‘(𝐷 FuncCat 𝐹))
57		eqid 2729	. . 3 ⊢ (oppCat‘(𝐵 FuncCat 𝐹)) = (oppCat‘(𝐵 FuncCat 𝐹))
58	54, 55, 6, 8, 13, 56, 57	ranfval 49596	. 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 2774	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ Ran 𝐸) = (⟨𝐵, 𝐷⟩ Ran 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4591  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  Catccat 17605  Hom_f chomf 17607  comp_fccomf 17608  oppCatcoppc 17652   Func cfunc 17796   FuncCat cfuc 17887   oppFunc coppf 49104   UP cup 49155   −∘_F cprcof 49355   Ran cran 49588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-ixp 8848  df-cat 17609  df-cid 17610  df-homf 17611  df-comf 17612  df-func 17800  df-nat 17888  df-fuc 17889  df-prcof 49356  df-ran 49590

This theorem is referenced by: (None)