Theorem prcofvala 50214

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)
prcofvala.f	⊢ (𝜑 → 𝐹 ∈ 𝑈)

Assertion

Ref	Expression
prcofvala	⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

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

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

Proof of Theorem prcofvala

Step	Hyp	Ref	Expression
1		prcofvalg.b	. 2 ⊢ 𝐵 = (𝐷 Func 𝐸)
2		prcofvalg.n	. 2 ⊢ 𝑁 = (𝐷 Nat 𝐸)
3		prcofvala.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
4		opex 5447	. . 3 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
5	4	a1i 11	. 2 ⊢ (𝜑 → ⟨𝐷, 𝐸⟩ ∈ V)
6		prcofvala.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
7		prcofvala.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
8		op1stg 8004	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
9	6, 7, 8	syl2anc 596	. 2 ⊢ (𝜑 → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
10		op2ndg 8005	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
11	6, 7, 10	syl2anc 596	. 2 ⊢ (𝜑 → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
12	1, 2, 3, 5, 9, 11	prcofvalg 50213	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   ↦ cmpt 5194   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991   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:  prcofval  50215  prcofpropd  50216  prcof1  50225  prcof2a  50226