Theorem 2arwcat 50707

Description: The condition for a structure with at most one object and at most two morphisms being a category. "2arwcat.2" to "2arwcat.5" are also necessary conditions if 𝑋, 0, and 1 are all sets, due to catlid 17857, catrid 17858, and catcocl 17859. (Contributed by Zhi Wang, 5-Nov-2025.)

Hypotheses

Ref	Expression
2arwcat.b	⊢ (𝜑 → {𝑋} = (Base‘𝐶))
2arwcat.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
2arwcat.x	⊢ (𝜑 → · = (comp‘𝐶))
2arwcat.1	⊢ (𝑋𝐻𝑋) = { 0 , 1 }
2arwcat.2	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )
2arwcat.3	⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )
2arwcat.4	⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )
2arwcat.5	⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })

Assertion

Ref	Expression
2arwcat	⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))

Distinct variable groups:   𝑦, ·   𝑦,𝐶   𝑦,𝐻   𝑦,𝑋   𝜑,𝑦

Allowed substitution hints:   1 (𝑦)   0 (𝑦)

Proof of Theorem 2arwcat

Dummy variables 𝑓 𝑔 𝑘 𝑤 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2arwcat.b	. 2 ⊢ (𝜑 → {𝑋} = (Base‘𝐶))
2		2arwcat.h	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
3		2arwcat.x	. 2 ⊢ (𝜑 → · = (comp‘𝐶))
4		2arwcat.2	. . . . . . 7 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )
5		ovex 7453	. . . . . . 7 ⊢ ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) ∈ V
6	4, 5	eqeltrrdi 2870	. . . . . 6 ⊢ (𝜑 → 1 ∈ V)
7		prid2g 4722	. . . . . 6 ⊢ ( 1 ∈ V → 1 ∈ { 0 , 1 })
8	6, 7	syl 18	. . . . 5 ⊢ (𝜑 → 1 ∈ { 0 , 1 })
9		2arwcat.1	. . . . 5 ⊢ (𝑋𝐻𝑋) = { 0 , 1 }
10	8, 9	eleqtrrdi 2872	. . . 4 ⊢ (𝜑 → 1 ∈ (𝑋𝐻𝑋))
11		df-ov 7423	. . . . 5 ⊢ (𝑋𝐻𝑋) = (𝐻‘⟨𝑋, 𝑋⟩)
12	2	fveq1d 6887	. . . . 5 ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑋⟩) = ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩))
13	11, 12	eqtrid 2808	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑋) = ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩))
14	10, 13	eleqtrd 2863	. . 3 ⊢ (𝜑 → 1 ∈ ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩))
15		elfv2ex 6928	. . 3 ⊢ ( 1 ∈ ((Hom ‘𝐶)‘⟨𝑋, 𝑋⟩) → 𝐶 ∈ V)
16	14, 15	syl 18	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
17	9	2arwcatlem1 50702	. 2 ⊢ ((((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 ))) ↔ ((𝑥 ∈ {𝑋} ∧ 𝑦 ∈ {𝑋}) ∧ (𝑧 ∈ {𝑋} ∧ 𝑤 ∈ {𝑋}) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
18	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑋}) → 1 ∈ { 0 , 1 })
19		velsn 4600	. . . . 5 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
20		id 23	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
21	20, 20	oveq12d 7438	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦𝐻𝑦) = (𝑋𝐻𝑋))
22	21, 9	eqtrdi 2812	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦𝐻𝑦) = { 0 , 1 })
23	19, 22	sylbi 220	. . . 4 ⊢ (𝑦 ∈ {𝑋} → (𝑦𝐻𝑦) = { 0 , 1 })
24	23	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑋}) → (𝑦𝐻𝑦) = { 0 , 1 })
25	18, 24	eleqtrrd 2864	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑋}) → 1 ∈ (𝑦𝐻𝑦))
26		simprll 791	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑥 = 𝑋 ∧ 𝑦 = 𝑋))
27	26	simpld 500	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑥 = 𝑋)
28	26	simprd 501	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑦 = 𝑋)
29		simprr1 1240	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑓 = 0 ∨ 𝑓 = 1 ))
30	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 1 )
31		2arwcat.3	. . . 4 ⊢ (𝜑 → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )
32	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = 0 )
33	27, 28, 28, 29, 30, 32	2arwcatlem2 50703	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
34		simprlr 792	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑧 = 𝑋 ∧ 𝑤 = 𝑋))
35	34	simpld 500	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑧 = 𝑋)
36		simprr2 1241	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔 = 0 ∨ 𝑔 = 1 ))
37		2arwcat.4	. . . 4 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )
38	37	adantr 486	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 0 )
39	28, 28, 35, 36, 30, 38	2arwcatlem3 50704	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧) 1 ) = 𝑔)
40		2arwcat.5	. . . . 5 ⊢ (𝜑 → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })
41	40	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ) ∈ { 0 , 1 })
42	27, 28, 35, 29, 30, 38, 32, 41, 36	2arwcatlem4 50705	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ { 0 , 1 })
43	27, 35	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑥𝐻𝑧) = (𝑋𝐻𝑋))
44	43, 9	eqtrdi 2812	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑥𝐻𝑧) = { 0 , 1 })
45	42, 44	eleqtrrd 2864	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
46	31, 37, 40	2arwcatlem5 50706	. . . . . . . 8 ⊢ (𝜑 → (( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )(⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋)( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )))
47	46	ad4antr 745	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )(⟨𝑋, 𝑋⟩ · 𝑋) 0 ) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋)( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )))
48		simpr 490	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → 𝑘 = 0 )
49		simplr 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → 𝑔 = 0 )
50	48, 49	oveq12d 7438	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ))
51		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) → 𝑓 = 0 )
52	51	ad2antrr 739	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → 𝑓 = 0 )
53	50, 52	oveq12d 7438	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )(⟨𝑋, 𝑋⟩ · 𝑋) 0 ))
54	49, 52	oveq12d 7438	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 ))
55	48, 54	oveq12d 7438	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = ( 0 (⟨𝑋, 𝑋⟩ · 𝑋)( 0 (⟨𝑋, 𝑋⟩ · 𝑋) 0 )))
56	47, 53, 55	3eqtr4d 2806	. . . . . 6 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
57		eqidd 2762	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑋 = 𝑋)
58	27, 28	opeq12d 4841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑋⟩)
59	58, 35	oveq12d 7438	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑋⟩ · 𝑋))
60	59	oveqd 7437	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
61	60, 42	eqeltrrd 2862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) ∈ { 0 , 1 })
62		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) ∈ V
63	62	elpr 4609	. . . . . . . . . 10 ⊢ ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) ∈ { 0 , 1 } ↔ ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 0 ∨ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 1 ))
64	61, 63	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 0 ∨ (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 1 ))
65	57, 57, 57, 64, 30, 32	2arwcatlem2 50703	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
66	65	ad3antrrr 743	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
67		simpr 490	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → 𝑘 = 1 )
68	67	oveq1d 7435	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
69	67	oveq1d 7435	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
70	57, 57, 57, 36, 30, 32	2arwcatlem2 50703	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑔)
71	70	ad3antrrr 743	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑔)
72	69, 71	eqtrd 2796	. . . . . . . 8 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑔)
73	72	oveq1d 7435	. . . . . . 7 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
74	66, 68, 73	3eqtr4rd 2807	. . . . . 6 ⊢ (((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) ∧ 𝑘 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
75		simprr3 1242	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘 = 0 ∨ 𝑘 = 1 ))
76	75	ad2antrr 739	. . . . . 6 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) → (𝑘 = 0 ∨ 𝑘 = 1 ))
77	56, 74, 76	mpjaodan 973	. . . . 5 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
78		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → 𝑔 = 1 )
79	78	oveq2d 7436	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋) 1 ))
80	57, 57, 57, 75, 30, 38	2arwcatlem3 50704	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑘)
81	80	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑘)
82	79, 81	eqtrd 2796	. . . . . . 7 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 𝑘)
83	82	oveq1d 7435	. . . . . 6 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
84	78	oveq1d 7435	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
85	57, 57, 57, 29, 30, 32	2arwcatlem2 50703	. . . . . . . . 9 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
86	85	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → ( 1 (⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
87	84, 86	eqtrd 2796	. . . . . . 7 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑓)
88	87	oveq2d 7436	. . . . . 6 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
89	83, 88	eqtr4d 2799	. . . . 5 ⊢ ((((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) ∧ 𝑔 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
90	36	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) → (𝑔 = 0 ∨ 𝑔 = 1 ))
91	77, 89, 90	mpjaodan 973	. . . 4 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 0 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
92	57, 57, 57, 36, 30, 38, 32, 41, 75	2arwcatlem4 50705	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ { 0 , 1 })
93		ovex 7453	. . . . . . . . 9 ⊢ (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ V
94	93	elpr 4609	. . . . . . . 8 ⊢ ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) ∈ { 0 , 1 } ↔ ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 0 ∨ (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 1 ))
95	92, 94	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 0 ∨ (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔) = 1 ))
96	57, 57, 57, 95, 30, 38	2arwcatlem3 50704	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
97	96	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
98		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → 𝑓 = 1 )
99	98	oveq2d 7436	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋) 1 ))
100	98	oveq2d 7436	. . . . . . 7 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑔(⟨𝑋, 𝑋⟩ · 𝑋) 1 ))
101	57, 57, 57, 36, 30, 38	2arwcatlem3 50704	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑔)
102	101	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋) 1 ) = 𝑔)
103	100, 102	eqtrd 2796	. . . . . 6 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = 𝑔)
104	103	oveq2d 7436	. . . . 5 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
105	97, 99, 104	3eqtr4d 2806	. . . 4 ⊢ (((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) ∧ 𝑓 = 1 ) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
106	91, 105, 29	mpjaodan 973	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
107	34	simprd 501	. . . . 5 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑤 = 𝑋)
108	58, 107	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑥, 𝑦⟩ · 𝑤) = (⟨𝑋, 𝑋⟩ · 𝑋))
109	28, 35	opeq12d 4841	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ⟨𝑦, 𝑧⟩ = ⟨𝑋, 𝑋⟩)
110	109, 107	oveq12d 7438	. . . . 5 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑦, 𝑧⟩ · 𝑤) = (⟨𝑋, 𝑋⟩ · 𝑋))
111	110	oveqd 7437	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔))
112		eqidd 2762	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑓 = 𝑓)
113	108, 111, 112	oveq123d 7441	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝑘(⟨𝑋, 𝑋⟩ · 𝑋)𝑔)(⟨𝑋, 𝑋⟩ · 𝑋)𝑓))
114	27, 35	opeq12d 4841	. . . . 5 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ⟨𝑥, 𝑧⟩ = ⟨𝑋, 𝑋⟩)
115	114, 107	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (⟨𝑥, 𝑧⟩ · 𝑤) = (⟨𝑋, 𝑋⟩ · 𝑋))
116		eqidd 2762	. . . 4 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → 𝑘 = 𝑘)
117	115, 116, 60	oveq123d 7441	. . 3 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) = (𝑘(⟨𝑋, 𝑋⟩ · 𝑋)(𝑔(⟨𝑋, 𝑋⟩ · 𝑋)𝑓)))
118	106, 113, 117	3eqtr4d 2806	. 2 ⊢ ((𝜑 ∧ (((𝑥 = 𝑋 ∧ 𝑦 = 𝑋) ∧ (𝑧 = 𝑋 ∧ 𝑤 = 𝑋)) ∧ ((𝑓 = 0 ∨ 𝑓 = 1 ) ∧ (𝑔 = 0 ∨ 𝑔 = 1 ) ∧ (𝑘 = 0 ∨ 𝑘 = 1 )))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
119	1, 2, 3, 16, 17, 25, 33, 39, 45, 118	iscatd2 17855	1 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ {𝑋} ↦ 1 )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  {cpr 4586  ⟨cop 4590   ↦ cmpt 5186  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439  compcco 17440  Catccat 17838  Idccid 17839

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-cat 17842  df-cid 17843

This theorem is used by:  incat  50708