Theorem prcof1 50854

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 486	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)
3		eqid 2763	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (𝐷 Func 𝐸)
4		eqid 2763	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
5		prcof1.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐾 ∈ (𝐷 Func 𝐸))
7	6	func1st2nd 50521	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
8	7	funcrcl2 50524	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐷 ∈ Cat)
9	7	funcrcl3 50525	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐸 ∈ Cat)
10		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
11	3, 4, 8, 9, 10	prcofvala 50843	. . . . . 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 6920	. . . . 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 7484	. . . . . . 7 ⊢ (𝐷 Func 𝐸) ∈ V
14	13	mptex 7260	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
15	13, 13	mpoex 8127	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
16	14, 15	op1st 8043	. . . . 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 2814	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
18	2, 17	eqtr3d 2800	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝑂 = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
19		simpr 490	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
20	19	oveq1d 7466	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → (𝑘 ∘func 𝐹) = (𝐾 ∘func 𝐹))
21		ovexd 7486	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) ∈ V)
22	18, 20, 6, 21	fvmptd 7032	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
23		0fv 6957	. . 3 ⊢ (∅‘𝐾) = ∅
24		reldmprcof 50841	. . . . . . . 8 ⊢ Rel dom −∘F
25	24	ovprc2 7491	. . . . . . 7 ⊢ (¬ 𝐹 ∈ V → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ∅)
26	25	fveq2d 6920	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘∅))
27		1st0 8041	. . . . . 6 ⊢ (1st ‘∅) = ∅
28	26, 27	eqtrdi 2814	. . . . 5 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = ∅)
29	1, 28	sylan9req 2819	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝑂 = ∅)
30	29	fveq1d 6918	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (∅‘𝐾))
31		df-cofu 18085	. . . . . 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 7584	. . . . 5 ⊢ Rel dom ∘func
33	32	ovprc2 7491	. . . 4 ⊢ (¬ 𝐹 ∈ V → (𝐾 ∘func 𝐹) = ∅)
34	33	adantl 487	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) = ∅)
35	23, 30, 34	3eqtr4a 2824	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
36	22, 35	pm2.61dan 825	1 ⊢ (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ⟨cop 4840   ↦ cmpt 5192  dom cdm 5658   ∘ ccom 5662  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  Catccat 17888   Func cfunc 18079   ∘_func ccofu 18081   Nat cnat 18169   −∘_F cprcof 50839

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-func 18083  df-cofu 18085  df-prcof 50840

This theorem is used by:  prcofdiag  50860  lanrcl5  51149  ranrcl5  51154  lanup  51155  ranup  51156