Theorem prcof1 50009

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 484	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)
3		eqid 2762	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (𝐷 Func 𝐸)
4		eqid 2762	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
5		prcof1.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	5	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐾 ∈ (𝐷 Func 𝐸))
7	6	func1st2nd 49697	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
8	7	funcrcl2 49700	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐷 ∈ Cat)
9	7	funcrcl3 49701	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐸 ∈ Cat)
10		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
11	3, 4, 8, 9, 10	prcofvala 49998	. . . . . 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 6871	. . . . 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 7429	. . . . . . 7 ⊢ (𝐷 Func 𝐸) ∈ V
14	13	mptex 7207	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
15	13, 13	mpoex 8060	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
16	14, 15	op1st 7978	. . . . 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 2813	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
18	2, 17	eqtr3d 2799	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝑂 = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
19		simpr 488	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
20	19	oveq1d 7411	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → (𝑘 ∘func 𝐹) = (𝐾 ∘func 𝐹))
21		ovexd 7431	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) ∈ V)
22	18, 20, 6, 21	fvmptd 6983	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
23		0fv 6908	. . 3 ⊢ (∅‘𝐾) = ∅
24		reldmprcof 49996	. . . . . . . 8 ⊢ Rel dom −∘F
25	24	ovprc2 7436	. . . . . . 7 ⊢ (¬ 𝐹 ∈ V → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ∅)
26	25	fveq2d 6871	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘∅))
27		1st0 7976	. . . . . 6 ⊢ (1st ‘∅) = ∅
28	26, 27	eqtrdi 2813	. . . . 5 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = ∅)
29	1, 28	sylan9req 2818	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝑂 = ∅)
30	29	fveq1d 6869	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (∅‘𝐾))
31		df-cofu 17893	. . . . . 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 7530	. . . . 5 ⊢ Rel dom ∘func
33	32	ovprc2 7436	. . . 4 ⊢ (¬ 𝐹 ∈ V → (𝐾 ∘func 𝐹) = ∅)
34	33	adantl 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) = ∅)
35	23, 30, 34	3eqtr4a 2823	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
36	22, 35	pm2.61dan 822	1 ⊢ (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  ⟨cop 4588   ↦ cmpt 5181  dom cdm 5647   ∘ ccom 5651  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  1^st c1st 7968  2^nd c2nd 7969  Catccat 17696   Func cfunc 17887   ∘_func ccofu 17889   Nat cnat 17977   −∘_F cprcof 49994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-func 17891  df-cofu 17893  df-prcof 49995

This theorem is referenced by:  prcofdiag  50015  lanrcl5  50256  ranrcl5  50261  lanup  50262  ranup  50263