Theorem prcofvala 49852

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 5416	. . 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 7954	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
9	6, 7, 8	syl2anc 585	. 2 ⊢ (𝜑 → (1st ‘⟨𝐷, 𝐸⟩) = 𝐷)
10		op2ndg 7955	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
11	6, 7, 10	syl2anc 585	. 2 ⊢ (𝜑 → (2nd ‘⟨𝐷, 𝐸⟩) = 𝐸)
12	1, 2, 3, 5, 9, 11	prcofvalg 49851	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 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   ↦ cmpt 5166   ∘ ccom 5635  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941   Func cfunc 17821   ∘_func ccofu 17823   Nat cnat 17911   −∘_F cprcof 49848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-prcof 49849

This theorem is referenced by:  prcofval  49853  prcofpropd  49854  prcof1  49863  prcof2a  49864