Theorem prcofvala 50040

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 5446	. . 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 7998	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
9	6, 7, 8	syl2anc 595	. 2 ⊢ (𝜑 → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
10		op2ndg 7999	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
11	6, 7, 10	syl2anc 595	. 2 ⊢ (𝜑 → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
12	1, 2, 3, 5, 9, 11	prcofvalg 50039	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   ↦ cmpt 5196   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985   Func cfunc 17911   ∘_func ccofu 17913   Nat cnat 18001   −∘_F cprcof 50036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-prcof 50037

This theorem is referenced by:  prcofval  50041  prcofpropd  50042  prcof1  50051  prcof2a  50052