Theorem prcof2a 49384

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 2729	. . . . . 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 49071	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
6	5	funcrcl2 49074	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49075	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		prcof2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
9	2, 3, 6, 7, 8	prcofvala 49372	. . . . 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 6826	. . . 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 7382	. . . . . 6 ⊢ (𝐷 Func 𝐸) ∈ V
12	11	mptex 7159	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
13	11, 11	mpoex 8014	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
14	12, 13	op2nd 7933	. . . 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 2780	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
16	1, 15	eqtr3d 2766	. 2 ⊢ (𝜑 → 𝑃 = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
17		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑘 = 𝐾)
18		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑙 = 𝐿)
19	17, 18	oveq12d 7367	. . 3 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑘𝑁𝑙) = (𝐾𝑁𝐿))
20	19	mpteq1d 5182	. 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 7382	. . . 4 ⊢ (𝐾𝑁𝐿) ∈ V
23	22	mptex 7159	. . 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 7501	1 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ (1st ‘𝐹))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ⟨cop 4583   ↦ cmpt 5173   ∘ ccom 5623  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923  Catccat 17570   Func cfunc 17761   ∘_func ccofu 17763   Nat cnat 17851   −∘_F cprcof 49368

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-func 17765  df-prcof 49369

This theorem is referenced by:  prcof21a  49386