Theorem prcofpropd 49358

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 17870	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	9	mpteq1d 5199	. . 3 ⊢ (𝜑 → (𝑘 ∈ (𝐴 Func 𝐶) ↦ (𝑘 ∘func 𝐹)) = (𝑘 ∈ (𝐵 Func 𝐷) ↦ (𝑘 ∘func 𝐹)))
11	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 Func 𝐶)) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
12	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (Homf ‘𝐴) = (Homf ‘𝐵))
13	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (compf‘𝐴) = (compf‘𝐵))
14	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (Homf ‘𝐶) = (Homf ‘𝐷))
15	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (compf‘𝐶) = (compf‘𝐷))
16		funcrcl 17831	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
17	16	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
18	17	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐴 ∈ Cat)
19	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐵 ∈ 𝑉)
20	12, 13, 18, 19	catpropd 17676	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 ∈ Cat ↔ 𝐵 ∈ Cat))
21	18, 20	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐵 ∈ Cat)
22	17	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐶 ∈ Cat)
23	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐷 ∈ 𝑉)
24	14, 15, 22, 23	catpropd 17676	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐶 ∈ Cat ↔ 𝐷 ∈ Cat))
25	22, 24	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → 𝐷 ∈ Cat)
26	12, 13, 14, 15, 18, 21, 22, 25	natpropd 17947	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
27	26	oveqd 7406	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝑘(𝐴 Nat 𝐶)𝑙) = (𝑘(𝐵 Nat 𝐷)𝑙))
28	27	mpteq1d 5199	. . . 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 7465	. . 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 4847	. 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 2730	. . 3 ⊢ (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
32		eqid 2730	. . 3 ⊢ (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
33		prcofpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
34	31, 32, 5, 7, 33	prcofvala 49356	. 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 2730	. . 3 ⊢ (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
36		eqid 2730	. . 3 ⊢ (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
37	35, 36, 6, 8, 33	prcofvala 49356	. 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 2775	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = (⟨𝐵, 𝐷⟩ −∘F 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4597   ↦ cmpt 5190   ∘ ccom 5644  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  1^st c1st 7968  Catccat 17631  Hom_f chomf 17633  comp_fccomf 17634   Func cfunc 17822   ∘_func ccofu 17824   Nat cnat 17912   −∘_F cprcof 49352

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-map 8803  df-ixp 8873  df-cat 17635  df-cid 17636  df-homf 17637  df-comf 17638  df-func 17826  df-nat 17914  df-prcof 49353

This theorem is referenced by:  lanpropd  49594  ranpropd  49595