Theorem prcof2 50667

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 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 50332	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
6	5	funcrcl2 50335	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 50336	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		prcof2.r	. . . . . 6 ⊢ Rel 𝑅
9		prcof2.f	. . . . . 6 ⊢ (𝜑 → 𝐹𝑅𝐺)
10	2, 3, 6, 7, 8, 9	prcofval 50655	. . . . 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 6900	. . . 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 7463	. . . . . 6 ⊢ (𝐷 Func 𝐸) ∈ V
13	12	mptex 7239	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)) ∈ V
14	12, 12	mpoex 8103	. . . . 5 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))) ∈ V
15	13, 14	op2nd 8020	. . . 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 2814	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩)) = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
17	1, 16	eqtr3d 2800	. 2 ⊢ (𝜑 → 𝑃 = (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
18		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑘 = 𝐾)
19		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → 𝑙 = 𝐿)
20	18, 19	oveq12d 7448	. . 3 ⊢ ((𝜑 ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑘𝑁𝑙) = (𝐾𝑁𝐿))
21	20	mpteq1d 5198	. 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 7463	. . . 4 ⊢ (𝐾𝑁𝐿) ∈ V
24	23	mptex 7239	. . 3 ⊢ (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)) ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)) ∈ V)
26	17, 21, 4, 22, 25	ovmpod 7582	1 ⊢ (𝜑 → (𝐾𝑃𝐿) = (𝑎 ∈ (𝐾𝑁𝐿) ↦ (𝑎 ∘ 𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3454  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   ∘ ccom 5659  Rel wrel 5660  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8009  2^nd c2nd 8010  Catccat 17853   Func cfunc 18044   ∘_func ccofu 18046   Nat cnat 18134   −∘_F cprcof 50650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-func 18048  df-prcof 50651

This theorem is used by: (None)