Theorem prcofvalg 50650

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)
prcofvalg.f	⊢ (𝜑 → 𝐹 ∈ 𝑈)
prcofvalg.p	⊢ (𝜑 → 𝑃 ∈ 𝑉)
prcofvalg.d	⊢ (𝜑 → (1st ‘𝑃) = 𝐷)
prcofvalg.e	⊢ (𝜑 → (2nd ‘𝑃) = 𝐸)

Assertion

Ref	Expression
prcofvalg	⊢ (𝜑 → (𝑃 −∘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   𝑃,𝑎,𝑘,𝑙   𝜑,𝑎,𝑘,𝑙

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

Proof of Theorem prcofvalg

Dummy variables 𝑏 𝑑 𝑒 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prcof 50648	. . 3 ⊢ −∘F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩)
2	1	a1i 11	. 2 ⊢ (𝜑 → −∘F = (𝑝 ∈ V, 𝑓 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩))
3		fvexd 6908	. . 3 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑝) ∈ V)
4		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → 𝑝 = 𝑃)
5	4	fveq2d 6897	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑝) = (1st ‘𝑃))
6		prcofvalg.d	. . . . 5 ⊢ (𝜑 → (1st ‘𝑃) = 𝐷)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑃) = 𝐷)
8	5, 7	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑝) = 𝐷)
9		fvexd 6908	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑝) ∈ V)
10	4	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → 𝑝 = 𝑃)
11	10	fveq2d 6897	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑝) = (2nd ‘𝑃))
12		prcofvalg.e	. . . . . 6 ⊢ (𝜑 → (2nd ‘𝑃) = 𝐸)
13	12	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑃) = 𝐸)
14	11, 13	eqtrd 2798	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑝) = 𝐸)
15		ovexd 7462	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Func 𝑒) ∈ V)
16		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → 𝑑 = 𝐷)
17		simpr 490	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
18	16, 17	oveq12d 7445	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Func 𝑒) = (𝐷 Func 𝐸))
19		prcofvalg.b	. . . . . 6 ⊢ 𝐵 = (𝐷 Func 𝐸)
20	18, 19	eqtr4di 2816	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Func 𝑒) = 𝐵)
21		simpr 490	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
22		simp-4r 796	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑝 = 𝑃 ∧ 𝑓 = 𝐹))
23	22	simprd 501	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
24	23	oveq2d 7443	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑘 ∘func 𝑓) = (𝑘 ∘func 𝐹))
25	21, 24	mpteq12dv 5194	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)) = (𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)))
26	16, 17	oveq12d 7445	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Nat 𝑒) = (𝐷 Nat 𝐸))
27		prcofvalg.n	. . . . . . . . . 10 ⊢ 𝑁 = (𝐷 Nat 𝐸)
28	26, 27	eqtr4di 2816	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Nat 𝑒) = 𝑁)
29	28	oveqdr 7455	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑘(𝑑 Nat 𝑒)𝑙) = (𝑘𝑁𝑙))
30	23	fveq2d 6897	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (1st ‘𝑓) = (1st ‘𝐹))
31	30	coeq2d 5844	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑎 ∘ (1st ‘𝑓)) = (𝑎 ∘ (1st ‘𝐹)))
32	29, 31	mpteq12dv 5194	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))) = (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))
33	21, 21, 32	mpoeq123dv 7502	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓)))) = (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))))
34	25, 33	opeq12d 4850	. . . . 5 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → ⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩ = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
35	15, 20, 34	csbied2 3887	. . . 4 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → ⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩ = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
36	9, 14, 35	csbied2 3887	. . 3 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → ⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩ = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
37	3, 8, 36	csbied2 3887	. 2 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → ⦋(1st ‘𝑝) / 𝑑⦌⦋(2nd ‘𝑝) / 𝑒⦌⦋(𝑑 Func 𝑒) / 𝑏⦌⟨(𝑘 ∈ 𝑏 ↦ (𝑘 ∘func 𝑓)), (𝑘 ∈ 𝑏, 𝑙 ∈ 𝑏 ↦ (𝑎 ∈ (𝑘(𝑑 Nat 𝑒)𝑙) ↦ (𝑎 ∘ (1st ‘𝑓))))⟩ = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
38		prcofvalg.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑉)
39	38	elexd 3476	. 2 ⊢ (𝜑 → 𝑃 ∈ V)
40		prcofvalg.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
41	40	elexd 3476	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
42		opex 5435	. . 3 ⊢ ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩ ∈ V
43	42	a1i 11	. 2 ⊢ (𝜑 → ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩ ∈ V)
44	2, 37, 39, 41, 43	ovmpod 7579	1 ⊢ (𝜑 → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ⦋csb 3850  ⟨cop 4837   ↦ cmpt 5188   ∘ ccom 5658  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  1^st c1st 8006  2^nd c2nd 8007   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

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-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-prcof 50648

This theorem is used by:  prcofvala  50651  prcofelvv  50654  reldmprcof1  50655  reldmprcof2  50656