Theorem prcofval 50652

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)
prcofval.r	⊢ Rel 𝑅
prcofval.f	⊢ (𝜑 → 𝐹𝑅𝐺)

Assertion

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

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

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

Proof of Theorem prcofval

Step	Hyp	Ref	Expression
1		prcofvalg.b	. . 3 ⊢ 𝐵 = (𝐷 Func 𝐸)
2		prcofvalg.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐸)
3		prcofvala.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
4		prcofvala.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
5		opex 5435	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
6	5	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
7	1, 2, 3, 4, 6	prcofvala 50651	. 2 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))))⟩)
8		prcofval.f	. . . . . . 7 ⊢ (𝜑 → 𝐹𝑅𝐺)
9		prcofval.r	. . . . . . . 8 ⊢ Rel 𝑅
10	9	brrelex12i 5709	. . . . . . 7 ⊢ (𝐹𝑅𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
11		op1stg 8020	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
12	8, 10, 11	3syl 19	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
13	12	coeq2d 5844	. . . . 5 ⊢ (𝜑 → (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝑎 ∘ 𝐹))
14	13	mpteq2dv 5201	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))) = (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))
15	14	mpoeq3dv 7506	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)))) = (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
16	15	opeq2d 4849	. 2 ⊢ (𝜑 → ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))))⟩ = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)
17	7, 16	eqtrd 2798	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ⟨cop 4837   class class class wbr 5105   ↦ cmpt 5188   ∘ ccom 5658  Rel wrel 5659  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  1^st c1st 8006   Func cfunc 18041   ∘_func ccofu 18043   Nat cnat 18131   −∘_F cprcof 50647

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-prcof 50648

This theorem is used by:  prcoftposcurfuco  50657  prcof2  50664