Theorem prcof2a 50052

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 2769	. . . . . 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 49739	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
6	5	funcrcl2 49742	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49743	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		prcof2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
9	2, 3, 6, 7, 8	prcofvala 50040	. . . . 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 6886	. . . 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 7444	. . . . . 6 ⊢ (𝐷 Func 𝐸) ∈ V
12	11	mptex 7222	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
13	11, 11	mpoex 8076	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
14	12, 13	op2nd 7995	. . . 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 2820	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
16	1, 15	eqtr3d 2806	. 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 7429	. . 3 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑘𝑁𝑙) = (𝐾𝑁𝐿))
20	19	mpteq1d 5205	. 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 7444	. . . 4 ⊢ (𝐾𝑁𝐿) ∈ V
23	22	mptex 7222	. . 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 7563	1 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   ↦ cmpt 5196   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  Catccat 17720   Func cfunc 17911   ∘_func ccofu 17913   Nat cnat 18001   −∘_F cprcof 50036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-func 17915  df-prcof 50037

This theorem is referenced by:  prcof21a  50054