Theorem prcofvala 49632

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 5412	. . 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 7945	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
9	6, 7, 8	syl2anc 584	. 2 ⊢ (𝜑 → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
10		op2ndg 7946	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
11	6, 7, 10	syl2anc 584	. 2 ⊢ (𝜑 → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
12	1, 2, 3, 5, 9, 11	prcofvalg 49631	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 1541   ∈ wcel 2113  Vcvv 3440  ⟨cop 4586   ↦ cmpt 5179   ∘ ccom 5628  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932   Func cfunc 17778   ∘_func ccofu 17780   Nat cnat 17868   −∘_F cprcof 49628

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-prcof 49629

This theorem is referenced by:  prcofval  49633  prcofpropd  49634  prcof1  49643  prcof2a  49644