Theorem prcofpropd 50216

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

Hypotheses

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

Assertion

Ref	Expression
prcofpropd	⊢ (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = (⟨𝐵, 𝐷⟩ −∘F 𝐹))

Proof of Theorem prcofpropd

Dummy variables 𝑎 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prcofpropd.1	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
2		prcofpropd.2	. . . . 5 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
3		prcofpropd.3	. . . . 5 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
4		prcofpropd.4	. . . . 5 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
5		prcofpropd.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		prcofpropd.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		prcofpropd.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
8		prcofpropd.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
9	1, 2, 3, 4, 5, 6, 7, 8	funcpropd 17983	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	9	mpteq1d 5203	. . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 Func 𝐶) ↦ (𝑘 ∘func 𝐹)) = (𝑘 ∈ (𝐵 Func 𝐷) ↦ (𝑘 ∘func 𝐹)))
11	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
12	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (Homf ‘𝐴) = (Homf ‘𝐵))
13	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (compf‘𝐴) = (compf‘𝐵))
14	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (compf‘𝐶) = (compf‘𝐷))
16		funcrcl 17944	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
17	16	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
18	17	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐴 ∈ Cat)
19	6	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐵 ∈ 𝑉)
20	12, 13, 18, 19	catpropd 17789	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 ∈ Cat ↔ 𝐵 ∈ Cat))
21	18, 20	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐵 ∈ Cat)
22	17	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐶 ∈ Cat)
23	8	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐷 ∈ 𝑉)
24	14, 15, 22, 23	catpropd 17789	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
25	22, 24	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐷 ∈ Cat)
26	12, 13, 14, 15, 18, 21, 22, 25	natpropd 18060	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
27	26	oveqd 7436	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝑘(𝐴 Nat 𝐶)𝑙) = (𝑘(𝐵 Nat 𝐷)𝑙))
28	27	mpteq1d 5203	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝑎 ∈ (𝑘(𝐴 Nat 𝐶)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))) = (𝑎 ∈ (𝑘(𝐵 Nat 𝐷)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))
29	9, 11, 28	mpoeq123dva 7493	. . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 Func 𝐶), 𝑙 ∈ (𝐴 Func 𝐶) ↦ (𝑎 ∈ (𝑘(𝐴 Nat 𝐶)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) = (𝑘 ∈ (𝐵 Func 𝐷), 𝑙 ∈ (𝐵 Func 𝐷) ↦ (𝑎 ∈ (𝑘(𝐵 Nat 𝐷)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
30	10, 29	opeq12d 4848	. 2 ⊢ (𝜑 → ⟨(𝑘 ∈ (𝐴 Func 𝐶) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐴 Func 𝐶), 𝑙 ∈ (𝐴 Func 𝐶) ↦ (𝑎 ∈ (𝑘(𝐴 Nat 𝐶)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩ = ⟨(𝑘 ∈ (𝐵 Func 𝐷) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐵 Func 𝐷), 𝑙 ∈ (𝐵 Func 𝐷) ↦ (𝑎 ∈ (𝑘(𝐵 Nat 𝐷)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
31		eqid 2765	. . 3 ⊢ (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
32		eqid 2765	. . 3 ⊢ (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
33		prcofpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
34	31, 32, 5, 7, 33	prcofvala 50214	. 2 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = ⟨(𝑘 ∈ (𝐴 Func 𝐶) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐴 Func 𝐶), 𝑙 ∈ (𝐴 Func 𝐶) ↦ (𝑎 ∈ (𝑘(𝐴 Nat 𝐶)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
35		eqid 2765	. . 3 ⊢ (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
36		eqid 2765	. . 3 ⊢ (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
37	35, 36, 6, 8, 33	prcofvala 50214	. 2 ⊢ (𝜑 → (⟨𝐵, 𝐷⟩ −∘F 𝐹) = ⟨(𝑘 ∈ (𝐵 Func 𝐷) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐵 Func 𝐷), 𝑙 ∈ (𝐵 Func 𝐷) ↦ (𝑎 ∈ (𝑘(𝐵 Nat 𝐷)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
38	30, 34, 37	3eqtr4d 2810	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = (⟨𝐵, 𝐷⟩ −∘F 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   ↦ cmpt 5194   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  Catccat 17744  Hom_f chomf 17746  comp_fccomf 17747   Func cfunc 17935   ∘_func ccofu 17937   Nat cnat 18025   −∘_F cprcof 50210

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-ixp 8902  df-cat 17748  df-cid 17749  df-homf 17750  df-comf 17751  df-func 17939  df-nat 18027  df-prcof 50211

This theorem is used by:  lanpropd  50452  ranpropd  50453