Theorem prcof2a 50187

Description: The morphism part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)

Hypotheses

Ref	Expression
prcof2a.n	⊢ 𝑁 = (𝐷 Nat 𝐸)
prcof2a.k	⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
prcof2a.l	⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
prcof2a.p	⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
prcof2a.f	⊢ (𝜑 → 𝐹 ∈ 𝑈)

Assertion

Ref	Expression
prcof2a	⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))

Distinct variable groups:   𝐷,𝑎   𝐸,𝑎   𝐹,𝑎   𝐾,𝑎   𝐿,𝑎   𝑁,𝑎   𝜑,𝑎

Allowed substitution hints:   𝑃(𝑎)   𝑈(𝑎)

Proof of Theorem prcof2a

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

Step	Hyp	Ref	Expression
1		prcof2a.p	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑃)
2		eqid 2763	. . . . . 6 ⊢ (𝐷 Func 𝐸) = (𝐷 Func 𝐸)
3		prcof2a.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐸)
4		prcof2a.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 49874	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
6	5	funcrcl2 49877	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49878	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		prcof2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
9	2, 3, 6, 7, 8	prcofvala 50175	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
10	9	fveq2d 6885	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (2nd ‘⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩))
11		ovex 7443	. . . . . 6 ⊢ (𝐷 Func 𝐸) ∈ V
12	11	mptex 7221	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
13	11, 11	mpoex 8072	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
14	12, 13	op2nd 7991	. . . 4 ⊢ (2nd ‘⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))
15	10, 14	eqtrdi 2814	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
16	1, 15	eqtr3d 2800	. 2 ⊢ (𝜑 → 𝑃 = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
17		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑘 = 𝐾)
18		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑙 = 𝐿)
19	17, 18	oveq12d 7428	. . 3 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑘𝑁𝑙) = (𝐾𝑁𝐿))
20	19	mpteq1d 5201	. 2 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))
21		prcof2a.l	. 2 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
22		ovex 7443	. . . 4 ⊢ (𝐾𝑁𝐿) ∈ V
23	22	mptex 7221	. . 3 ⊢ (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))) ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))) ∈ V)
25	16, 20, 4, 21, 24	ovmpod 7562	1 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   ↦ cmpt 5192   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  Catccat 17715   Func cfunc 17906   ∘_func ccofu 17908   Nat cnat 17996   −∘_F cprcof 50171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-func 17910  df-prcof 50172

This theorem is referenced by:  prcof21a  50189