Theorem isinito2lem 49467

Description: The predicate "is an initial object" of a category, using universal property. (Contributed by Zhi Wang, 23-Oct-2025.)

Hypotheses

Ref	Expression
isinito2.1	⊢ 1 = (SetCat‘1o)
isinito2.f	⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)
isinito2lem.c	⊢ (𝜑 → 𝐶 ∈ Cat)
isinito2lem.i	⊢ (𝜑 → 𝐼 ∈ (Base‘𝐶))

Assertion

Ref	Expression
isinito2lem	⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅))

Proof of Theorem isinito2lem

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reutru 48782	. . . . 5 ⊢ (∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤)
2		0ex 5264	. . . . . . . 8 ⊢ ∅ ∈ V
3		eqeq1 2734	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
4	3	reubidv 3374	. . . . . . . 8 ⊢ (𝑦 = ∅ → (∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
5	2, 4	ralsn 4647	. . . . . . 7 ⊢ (∀𝑦 ∈ {∅}∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅))
6		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ ( 1 Δfunc𝐶) = ( 1 Δfunc𝐶)
7		isinito2.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 = (SetCat‘1o)
8		setc1oterm 49460	. . . . . . . . . . . . . . . . . . . 20 ⊢ (SetCat‘1o) ∈ TermCat
9	7, 8	eqeltri 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ TermCat
10	9	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ TermCat)
11	10	termccd 49448	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ Cat)
12		isinito2lem.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	7	setc1obas 49461	. . . . . . . . . . . . . . . . 17 ⊢ 1o = (Base‘ 1 )
14		0lt1o 8470	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 1o
15	14	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 1o)
16		isinito2.f	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)
17		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		isinito2lem.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ (Base‘𝐶))
19	6, 11, 12, 13, 15, 16, 17, 18	diag11 18210	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1st ‘𝐹)‘𝐼) = ∅)
20	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝐼) = ∅)
21	20	opeq2d 4846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨∅, ((1st ‘𝐹)‘𝐼)⟩ = ⟨∅, ∅⟩)
22	11	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 1 ∈ Cat)
23	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
24	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∅ ∈ 1o)
25		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
26	6, 22, 23, 13, 24, 16, 17, 25	diag11 18210	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) = ∅)
27	21, 26	oveq12d 7407	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
28		snex 5393	. . . . . . . . . . . . . 14 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
29	28	ovsn2 48837	. . . . . . . . . . . . 13 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
30	27, 29	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = {⟨∅, ∅, ∅⟩})
31	30	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = {⟨∅, ∅, ∅⟩})
32	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 1 ∈ TermCat)
33	32	termccd 49448	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 1 ∈ Cat)
34	12	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝐶 ∈ Cat)
35	14	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ ∈ 1o)
36	18	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝐼 ∈ (Base‘𝐶))
37		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Id‘ 1 ) = (Id‘ 1 )
39		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
40		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥))
41	6, 33, 34, 13, 35, 16, 17, 36, 37, 38, 39, 40	diag12 18211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ((Id‘ 1 )‘∅))
42	7, 38	setc1oid 49464	. . . . . . . . . . . 12 ⊢ ((Id‘ 1 )‘∅) = ∅
43	41, 42	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ∅)
44		eqidd 2731	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = ∅)
45	31, 43, 44	oveq123d 7410	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) = (∅{⟨∅, ∅, ∅⟩}∅))
46	2	ovsn2 48837	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
47	45, 46	eqtr2di 2782	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅))
48		tbtru 1548	. . . . . . . . 9 ⊢ (∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ (∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ⊤))
49	47, 48	sylib 218	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ⊤))
50	49	reubidva 3372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤))
51	5, 50	bitr2id 284	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤ ↔ ∀𝑦 ∈ {∅}∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
52	26	oveq2d 7405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = (∅{⟨∅, ∅, 1o⟩}∅))
53		1oex 8446	. . . . . . . . . 10 ⊢ 1o ∈ V
54	53	ovsn2 48837	. . . . . . . . 9 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
55		df1o2 8443	. . . . . . . . 9 ⊢ 1o = {∅}
56	54, 55	eqtri 2753	. . . . . . . 8 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = {∅}
57	52, 56	eqtrdi 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = {∅})
58	57	raleqdv 3301	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑦 ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥))∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ∀𝑦 ∈ {∅}∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
59	51, 58	bitr4d 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤ ↔ ∀𝑦 ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥))∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
60	1, 59	bitrid 283	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥) ↔ ∀𝑦 ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥))∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
61	60	ralbidva 3155	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥))∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
62	17, 37, 12, 18	isinito 17964	. . 3 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑥 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)))
63	7	setc1ohomfval 49462	. . . 4 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
64	7	setc1ocofval 49463	. . . 4 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
65	7, 16, 12	funcsetc1ocl 49465	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))
66	65	func1st2nd 49053	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 1 )(2nd ‘𝐹))
67	19	oveq2d 7405	. . . . . 6 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = (∅{⟨∅, ∅, 1o⟩}∅))
68	67, 54	eqtrdi 2781	. . . . 5 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = 1o)
69	14, 68	eleqtrrid 2836	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)))
70	17, 13, 37, 63, 64, 15, 66, 18, 69	isup 49153	. . 3 ⊢ (𝜑 → (𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 1 )∅)∅ ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥))∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)𝑦 = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅)))
71	61, 62, 70	3bitr4d 311	. 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 1 )∅)∅))
72	65	up1st2ndb 49160	. 2 ⊢ (𝜑 → (𝐼(𝐹(𝐶 UP 1 )∅)∅ ↔ 𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 1 )∅)∅))
73	71, 72	bitr4d 282	1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ∃!weu 2562  ∀wral 3045  ∃!wreu 3354  ∅c0 4298  {csn 4591  ⟨cop 4597  ⟨cotp 4599   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  1_oc1o 8429  Basecbs 17185  Hom chom 17237  Catccat 17631  Idccid 17632  InitOcinito 17949  SetCatcsetc 18043  Δ_funccdiag 18179   UP cup 49146  TermCatctermc 49441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-func 17826  df-nat 17914  df-fuc 17915  df-inito 17952  df-setc 18044  df-xpc 18139  df-1stf 18140  df-curf 18181  df-diag 18183  df-up 49147  df-thinc 49387  df-termc 49442

This theorem is referenced by:  isinito2  49468