Theorem prcofpropd 49738

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 17838	. . . 4 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
10	9	mpteq1d 5190	. . 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 17799	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝐴 Func 𝐶) → (𝐴 ∈ Cat ∧ 𝐶 ∈ Cat))
17	16	ad2antrl 729	. . . . . . . 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 17644	. . . . . . . 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 17644	. . . . . . . 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 17915	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
27	26	oveqd 7385	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝐴 Func 𝐶) ∧ 𝑙 ∈ (𝐴 Func 𝐶))) → (𝑘(𝐴 Nat 𝐶)𝑙) = (𝑘(𝐵 Nat 𝐷)𝑙))
28	27	mpteq1d 5190	. . . 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 7442	. . 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 4839	. 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 2737	. . 3 ⊢ (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
32		eqid 2737	. . 3 ⊢ (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
33		prcofpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
34	31, 32, 5, 7, 33	prcofvala 49736	. 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 2737	. . 3 ⊢ (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
36		eqid 2737	. . 3 ⊢ (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
37	35, 36, 6, 8, 33	prcofvala 49736	. 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 2782	1 ⊢ (𝜑 → (⟨𝐴, 𝐶⟩ −∘F 𝐹) = (⟨𝐵, 𝐷⟩ −∘F 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   ↦ cmpt 5181   ∘ ccom 5636  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  Catccat 17599  Hom_f chomf 17601  comp_fccomf 17602   Func cfunc 17790   ∘_func ccofu 17792   Nat cnat 17880   −∘_F cprcof 49732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ixp 8848  df-cat 17603  df-cid 17604  df-homf 17605  df-comf 17606  df-func 17794  df-nat 17882  df-prcof 49733

This theorem is referenced by:  lanpropd  49974  ranpropd  49975