Theorem prcof1 50662

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 50329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
8	7	funcrcl2 50332	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐷 ∈ Cat)
9	7	funcrcl3 50333	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐸 ∈ Cat)
10		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
11	3, 4, 8, 9, 10	prcofvala 50651	. . . . . 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 6897	. . . . 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 7460	. . . . . . 7 ⊢ (𝐷 Func 𝐸) ∈ V
14	13	mptex 7236	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
15	13, 13	mpoex 8100	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
16	14, 15	op1st 8016	. . . . 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 7442	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → (𝑘 ∘func 𝐹) = (𝐾 ∘func 𝐹))
21		ovexd 7462	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) ∈ V)
22	18, 20, 6, 21	fvmptd 7009	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
23		0fv 6934	. . 3 ⊢ (∅‘𝐾) = ∅
24		reldmprcof 50649	. . . . . . . 8 ⊢ Rel dom −∘F
25	24	ovprc2 7467	. . . . . . 7 ⊢ (¬ 𝐹 ∈ V → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ∅)
26	25	fveq2d 6897	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘∅))
27		1st0 8014	. . . . . 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 6895	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (∅‘𝐾))
31		df-cofu 18047	. . . . . 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 7561	. . . . 5 ⊢ Rel dom ∘func
33	32	ovprc2 7467	. . . 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 1569   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  ⟨cop 4837   ↦ cmpt 5188  dom cdm 5654   ∘ ccom 5658  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429  1^st c1st 8006  2^nd c2nd 8007  Catccat 17850   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-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

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-reu 3369  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-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  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-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-func 18045  df-cofu 18047  df-prcof 50648

This theorem is used by:  prcofdiag  50668  lanrcl5  50957  ranrcl5  50962  lanup  50963  ranup  50964