Theorem prcof2 49372

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))
prcof2.p	⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = 𝑃)
prcof2.r	⊢ Rel 𝑅
prcof2.f	⊢ (𝜑 → 𝐹𝑅𝐺)

Assertion

Ref	Expression
prcof2	⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)))

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

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

Proof of Theorem prcof2

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

Step	Hyp	Ref	Expression
1		prcof2.p	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = 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 49058	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
6	5	funcrcl2 49061	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49062	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		prcof2.r	. . . . . 6 ⊢ Rel 𝑅
9		prcof2.f	. . . . . 6 ⊢ (𝜑 → 𝐹𝑅𝐺)
10	2, 3, 6, 7, 8, 9	prcofval 49360	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)
11	10	fveq2d 6844	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = (2nd ‘⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩))
12		ovex 7402	. . . . . 6 ⊢ (𝐷 Func 𝐸) ∈ V
13	12	mptex 7179	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)) ∈ V
14	12, 12	mpoex 8037	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))) ∈ V
15	13, 14	op2nd 7956	. . . 4 ⊢ (2nd ‘⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))
16	11, 15	eqtrdi 2780	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
17	1, 16	eqtr3d 2766	. 2 ⊢ (𝜑 → 𝑃 = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
18		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑘 = 𝐾)
19		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑙 = 𝐿)
20	18, 19	oveq12d 7387	. . 3 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑘𝑁𝑙) = (𝐾𝑁𝐿))
21	20	mpteq1d 5192	. 2 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)))
22		prcof2a.l	. 2 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
23		ovex 7402	. . . 4 ⊢ (𝐾𝑁𝐿) ∈ V
24	23	mptex 7179	. . 3 ⊢ (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)) ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)) ∈ V)
26	17, 21, 4, 22, 25	ovmpod 7521	1 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   class class class wbr 5102   ↦ cmpt 5183   ∘ ccom 5635  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  Catccat 17605   Func cfunc 17796   ∘_func ccofu 17798   Nat cnat 17886   −∘_F cprcof 49355

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-func 17800  df-prcof 49356

This theorem is referenced by: (None)