Theorem prcof1 50051

Description: The object part of the pre-composition functor. (Contributed by Zhi Wang, 3-Nov-2025.)

Hypotheses

Ref	Expression
prcof1.k	⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
prcof1.o	⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)

Assertion

Ref	Expression
prcof1	⊢ (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))

Proof of Theorem prcof1

Dummy variables 𝑎 𝑏 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prcof1.o	. . . . 5 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)
2	1	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)
3		eqid 2769	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (𝐷 Func 𝐸)
4		eqid 2769	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
5		prcof1.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	5	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐾 ∈ (𝐷 Func 𝐸))
7	6	func1st2nd 49739	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
8	7	funcrcl2 49742	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐷 ∈ Cat)
9	7	funcrcl3 49743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐸 ∈ Cat)
10		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
11	3, 4, 8, 9, 10	prcofvala 50040	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)
12	11	fveq2d 6886	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩))
13		ovex 7444	. . . . . . 7 ⊢ (𝐷 Func 𝐸) ∈ V
14	13	mptex 7222	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
15	13, 13	mpoex 8076	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
16	14, 15	op1st 7994	. . . . 5 ⊢ (1st ‘⟨(𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)), (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩) = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹))
17	12, 16	eqtrdi 2820	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
18	2, 17	eqtr3d 2806	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝑂 = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
19		simpr 489	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
20	19	oveq1d 7426	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → (𝑘 ∘func 𝐹) = (𝐾 ∘func 𝐹))
21		ovexd 7446	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) ∈ V)
22	18, 20, 6, 21	fvmptd 6998	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
23		0fv 6923	. . 3 ⊢ (∅‘𝐾) = ∅
24		reldmprcof 50038	. . . . . . . 8 ⊢ Rel dom −∘F
25	24	ovprc2 7451	. . . . . . 7 ⊢ (¬ 𝐹 ∈ V → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ∅)
26	25	fveq2d 6886	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘∅))
27		1st0 7992	. . . . . 6 ⊢ (1st ‘∅) = ∅
28	26, 27	eqtrdi 2820	. . . . 5 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = ∅)
29	1, 28	sylan9req 2825	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝑂 = ∅)
30	29	fveq1d 6884	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (∅‘𝐾))
31		df-cofu 17917	. . . . . 6 ⊢ ∘func = (𝑙 ∈ V, 𝑘 ∈ V ↦ ⟨((1st ‘𝑙) ∘ (1st ‘𝑘)), (𝑎 ∈ dom dom (2nd ‘𝑘), 𝑏 ∈ dom dom (2nd ‘𝑘) ↦ ((((1st ‘𝑘)‘𝑎)(2nd ‘𝑙)((1st ‘𝑘)‘𝑏)) ∘ (𝑎(2nd ‘𝑘)𝑏)))⟩)
32	31	reldmmpo 7545	. . . . 5 ⊢ Rel dom ∘func
33	32	ovprc2 7451	. . . 4 ⊢ (¬ 𝐹 ∈ V → (𝐾 ∘func 𝐹) = ∅)
34	33	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) = ∅)
35	23, 30, 34	3eqtr4a 2830	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
36	22, 35	pm2.61dan 824	1 ⊢ (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  ⟨cop 4600   ↦ cmpt 5196  dom cdm 5662   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  Catccat 17720   Func cfunc 17911   ∘_func ccofu 17913   Nat cnat 18001   −∘_F cprcof 50036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-func 17915  df-cofu 17917  df-prcof 50037

This theorem is referenced by:  prcofdiag  50057  lanrcl5  50298  ranrcl5  50303  lanup  50304  ranup  50305