Theorem prcofval 49853

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 5416	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
6	5	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
7	1, 2, 3, 4, 6	prcofvala 49852	. 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 5686	. . . . . . 7 ⊢ (𝐹𝑅𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
11		op1stg 7954	. . . . . . 7 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
12	8, 10, 11	3syl 18	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
13	12	coeq2d 5817	. . . . 5 ⊢ (𝜑 → (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)) = (𝑎 ∘ 𝐹))
14	13	mpteq2dv 5179	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩))) = (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))
15	14	mpoeq3dv 7446	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘⟨𝐹, 𝐺⟩)))) = (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹))))
16	15	opeq2d 4823	. 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 2771	1 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F ⟨𝐹, 𝐺⟩) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func ⟨𝐹, 𝐺⟩)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ 𝐹)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   class class class wbr 5085   ↦ cmpt 5166   ∘ ccom 5635  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940   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:  prcoftposcurfuco  49858  prcof2  49865