Theorem prcof1 49878

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 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = 𝑂)
3		eqid 2737	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (𝐷 Func 𝐸)
4		eqid 2737	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
5		prcof1.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐾 ∈ (𝐷 Func 𝐸))
7	6	func1st2nd 49566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
8	7	funcrcl2 49569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐷 ∈ Cat)
9	7	funcrcl3 49570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐸 ∈ Cat)
10		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝐹 ∈ V)
11	3, 4, 8, 9, 10	prcofvala 49867	. . . . . 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 6839	. . . . 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 7394	. . . . . . 7 ⊢ (𝐷 Func 𝐸) ∈ V
14	13	mptex 7172	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)) ∈ V
15	13, 13	mpoex 8026	. . . . . 6 ⊢ (𝑘 ∈ (𝐷 Func 𝐸), 𝑙 ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑘(𝐷 Nat 𝐸)𝑙) ↦ (𝑎 ∘ (1st ‘𝐹)))) ∈ V
16	14, 15	op1st 7944	. . . . 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 2788	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
18	2, 17	eqtr3d 2774	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → 𝑂 = (𝑘 ∈ (𝐷 Func 𝐸) ↦ (𝑘 ∘func 𝐹)))
19		simpr 484	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
20	19	oveq1d 7376	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ V) ∧ 𝑘 = 𝐾) → (𝑘 ∘func 𝐹) = (𝐾 ∘func 𝐹))
21		ovexd 7396	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) ∈ V)
22	18, 20, 6, 21	fvmptd 6950	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
23		0fv 6876	. . 3 ⊢ (∅‘𝐾) = ∅
24		reldmprcof 49865	. . . . . . . 8 ⊢ Rel dom −∘F
25	24	ovprc2 7401	. . . . . . 7 ⊢ (¬ 𝐹 ∈ V → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ∅)
26	25	fveq2d 6839	. . . . . 6 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘∅))
27		1st0 7942	. . . . . 6 ⊢ (1st ‘∅) = ∅
28	26, 27	eqtrdi 2788	. . . . 5 ⊢ (¬ 𝐹 ∈ V → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = ∅)
29	1, 28	sylan9req 2793	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → 𝑂 = ∅)
30	29	fveq1d 6837	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (∅‘𝐾))
31		df-cofu 17821	. . . . . 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 7495	. . . . 5 ⊢ Rel dom ∘func
33	32	ovprc2 7401	. . . 4 ⊢ (¬ 𝐹 ∈ V → (𝐾 ∘func 𝐹) = ∅)
34	33	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝐾 ∘func 𝐹) = ∅)
35	23, 30, 34	3eqtr4a 2798	. 2 ⊢ ((𝜑 ∧ ¬ 𝐹 ∈ V) → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))
36	22, 35	pm2.61dan 813	1 ⊢ (𝜑 → (𝑂‘𝐾) = (𝐾 ∘func 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ⟨cop 4574   ↦ cmpt 5167  dom cdm 5625   ∘ ccom 5629  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  1^st c1st 7934  2^nd c2nd 7935  Catccat 17624   Func cfunc 17815   ∘_func ccofu 17817   Nat cnat 17905   −∘_F cprcof 49863

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-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-func 17819  df-cofu 17821  df-prcof 49864

This theorem is referenced by:  prcofdiag  49884  lanrcl5  50125  ranrcl5  50130  lanup  50131  ranup  50132