Theorem prcofval 50215

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 5447	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
6	5	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
7	1, 2, 3, 4, 6	prcofvala 50214	. 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 5718	. . . . . . 7 ⊢ (𝐹𝑅𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
11		op1stg 8004	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
12	8, 10, 11	3syl 19	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
13	12	coeq2d 5850	. . . . 5 ⊢ (𝜑 → (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝑎 ∘ 𝐹))
14	13	mpteq2dv 5207	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))) = (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))
15	14	mpoeq3dv 7498	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)))) = (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
16	15	opeq2d 4847	. 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 2800	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   ∘ ccom 5667  Rel wrel 5668  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990   Func cfunc 17935   ∘_func ccofu 17937   Nat cnat 18025   −∘_F cprcof 50210

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-prcof 50211

This theorem is used by:  prcoftposcurfuco  50220  prcof2  50227