Theorem prcofvalg 49735

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 49733	. . 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 6857	. . 3 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑝) ∈ V)
4		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → 𝑝 = 𝑃)
5	4	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑝) = (1st ‘𝑃))
6		prcofvalg.d	. . . . 5 ⊢ (𝜑 → (1st ‘𝑃) = 𝐷)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑃) = 𝐷)
8	5, 7	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) → (1st ‘𝑝) = 𝐷)
9		fvexd 6857	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑝) ∈ V)
10	4	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → 𝑝 = 𝑃)
11	10	fveq2d 6846	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑝) = (2nd ‘𝑃))
12		prcofvalg.e	. . . . . 6 ⊢ (𝜑 → (2nd ‘𝑃) = 𝐸)
13	12	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑃) = 𝐸)
14	11, 13	eqtrd 2772	. . . 4 ⊢ (((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) → (2nd ‘𝑝) = 𝐸)
15		ovexd 7403	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Func 𝑒) ∈ V)
16		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → 𝑑 = 𝐷)
17		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
18	16, 17	oveq12d 7386	. . . . . 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 2790	. . . . 5 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Func 𝑒) = 𝐵)
21		simpr 484	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
22		simp-4r 784	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑝 = 𝑃 ∧ 𝑓 = 𝐹))
23	22	simprd 495	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → 𝑓 = 𝐹)
24	23	oveq2d 7384	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑘 ∘func 𝑓) = (𝑘 ∘func 𝐹))
25	21, 24	mpteq12dv 5187	. . . . . 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 7386	. . . . . . . . . 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 2790	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) → (𝑑 Nat 𝑒) = 𝑁)
29	28	oveqdr 7396	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑘(𝑑 Nat 𝑒)𝑙) = (𝑘𝑁𝑙))
30	23	fveq2d 6846	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (1st ‘𝑓) = (1st ‘𝐹))
31	30	coeq2d 5819	. . . . . . . 8 ⊢ (((((𝜑 ∧ (𝑝 = 𝑃 ∧ 𝑓 = 𝐹)) ∧ 𝑑 = 𝐷) ∧ 𝑒 = 𝐸) ∧ 𝑏 = 𝐵) → (𝑎 ∘ (1st ‘𝑓)) = (𝑎 ∘ (1st ‘𝐹)))
32	29, 31	mpteq12dv 5187	. . . . . . 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 7443	. . . . . 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 4839	. . . . 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 3888	. . . 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 3888	. . 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 3888	. 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 3466	. 2 ⊢ (𝜑 → 𝑃 ∈ V)
40		prcofvalg.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑈)
41	40	elexd 3466	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
42		opex 5419	. . 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 7520	1 ⊢ (𝜑 → (𝑃 −∘F 𝐹) = ⟨(𝑘 ∈ 𝐵 ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ 𝐵, 𝑙 ∈ 𝐵 ↦ (𝑎 ∈ (𝑘𝑁𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⦋csb 3851  ⟨cop 4588   ↦ cmpt 5181   ∘ ccom 5636  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942   Func cfunc 17790   ∘_func ccofu 17792   Nat cnat 17880   −∘_F cprcof 49732

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-prcof 49733

This theorem is referenced by:  prcofvala  49736  prcofelvv  49739  reldmprcof1  49740  reldmprcof2  49741