Theorem prcofval 49565

Description: Value of the pre-composition functor. (Contributed by Zhi Wang, 2-Nov-2025.)

Hypotheses

Ref	Expression
prcofvalg.b	⊢ 𝐵 = (𝐷 Func 𝐸)
prcofvalg.n	⊢ 𝑁 = (𝐷 Nat 𝐸)
prcofvala.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
prcofvala.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)
prcofval.r	⊢ Rel 𝑅
prcofval.f	⊢ (𝜑 → 𝐹𝑅𝐺)

Assertion

Ref	Expression
prcofval	⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)

Distinct variable groups:   𝐵,𝑎,𝑘,𝑙   𝐷,𝑎,𝑘,𝑙   𝐸,𝑎,𝑘,𝑙   𝐹,𝑎,𝑘,𝑙   𝐺,𝑎,𝑘,𝑙   𝜑,𝑎,𝑘,𝑙

Allowed substitution hints:   𝑅(𝑘,𝑎,𝑙)   𝑁(𝑘,𝑎,𝑙)   𝑉(𝑘,𝑎,𝑙)   𝑊(𝑘,𝑎,𝑙)

Proof of Theorem prcofval

Step	Hyp	Ref	Expression
1		prcofvalg.b	. . 3 ⊢ 𝐵 = (𝐷 Func 𝐸)
2		prcofvalg.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐸)
3		prcofvala.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
4		prcofvala.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
5		opex 5410	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
6	5	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
7	1, 2, 3, 4, 6	prcofvala 49564	. 2 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))))⟩)
8		prcofval.f	. . . . . . 7 ⊢ (𝜑 → 𝐹𝑅𝐺)
9		prcofval.r	. . . . . . . 8 ⊢ Rel 𝑅
10	9	brrelex12i 5677	. . . . . . 7 ⊢ (𝐹𝑅𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
11		op1stg 7943	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
12	8, 10, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
13	12	coeq2d 5809	. . . . 5 ⊢ (𝜑 → (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝑎 ∘ 𝐹))
14	13	mpteq2dv 5190	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))) = (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))
15	14	mpoeq3dv 7435	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)))) = (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
16	15	opeq2d 4834	. 2 ⊢ (𝜑 → ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))))⟩ = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)
17	7, 16	eqtrd 2769	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4584   class class class wbr 5096   ↦ cmpt 5177   ∘ ccom 5626  Rel wrel 5627  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929   Func cfunc 17776   ∘_func ccofu 17778   Nat cnat 17866   −∘_F cprcof 49560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-prcof 49561

This theorem is referenced by:  prcoftposcurfuco  49570  prcof2  49577