Theorem prcofvala 50401

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 5432	. . 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 7997	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
9	6, 7, 8	syl2anc 596	. 2 ⊢ (𝜑 → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
10		op2ndg 7998	. . 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 50400	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 2145  Vcvv 3450  ⟨cop 4590   ↦ cmpt 5186   ∘ ccom 5652  ‘cfv 6528  (class class class)co 7409   ∈ cmpo 7411  1^st c1st 7983  2^nd c2nd 7984   Func cfunc 17976   ∘_func ccofu 17978   Nat cnat 18066   −∘_F cprcof 50397

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6484  df-fun 6530  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7985  df-2nd 7986  df-prcof 50398

This theorem is used by:  prcofval  50402  prcofpropd  50403  prcof1  50412  prcof2a  50413