Theorem isinito2lem 49196

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(𝐶UP 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 48697	. . . . 5 ⊢ (∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤)
2		0ex 5287	. . . . . . . 8 ⊢ ∅ ∈ V
3		eqeq1 2738	. . . . . . . . 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 3381	. . . . . . . 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 4661	. . . . . . 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 2734	. . . . . . . . . . . . . . . . 17 ⊢ ( 1 Δfunc𝐶) = ( 1 Δfunc𝐶)
7		isinito2.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 = (SetCat‘1o)
8		setc1oterm 49189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (SetCat‘1o) ∈ TermCat
9	7, 8	eqeltri 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ TermCat
10	9	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ TermCat)
11	10	termccd 49178	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ Cat)
12		isinito2lem.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	7	setc1obas 49190	. . . . . . . . . . . . . . . . 17 ⊢ 1o = (Base‘ 1 )
14		0lt1o 8524	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 1o
15	14	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 1o)
16		isinito2.f	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)
17		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		isinito2lem.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ (Base‘𝐶))
19	6, 11, 12, 13, 15, 16, 17, 18	diag11 18259	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1st ‘𝐹)‘𝐼) = ∅)
20	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝐼) = ∅)
21	20	opeq2d 4860	. . . . . . . . . . . . . 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 18259	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) = ∅)
27	21, 26	oveq12d 7431	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
28		snex 5416	. . . . . . . . . . . . . 14 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
29	28	ovsn2 48745	. . . . . . . . . . . . 13 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
30	27, 29	eqtrdi 2785	. . . . . . . . . . . 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 49178	. . . . . . . . . . . . 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 2734	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38		eqid 2734	. . . . . . . . . . . . 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 18260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ((Id‘ 1 )‘∅))
42	7, 38	setc1oid 49193	. . . . . . . . . . . 12 ⊢ ((Id‘ 1 )‘∅) = ∅
43	41, 42	eqtrdi 2785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ∅)
44		eqidd 2735	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = ∅)
45	31, 43, 44	oveq123d 7434	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) = (∅{⟨∅, ∅, ∅⟩}∅))
46	2	ovsn2 48745	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
47	45, 46	eqtr2di 2786	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅))
48		tbtru 1547	. . . . . . . . 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 3379	. . . . . . 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 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = (∅{⟨∅, ∅, 1o⟩}∅))
53		1oex 8498	. . . . . . . . . 10 ⊢ 1o ∈ V
54	53	ovsn2 48745	. . . . . . . . 9 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
55		df1o2 8495	. . . . . . . . 9 ⊢ 1o = {∅}
56	54, 55	eqtri 2757	. . . . . . . 8 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = {∅}
57	52, 56	eqtrdi 2785	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = {∅})
58	57	raleqdv 3309	. . . . . 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 3163	. . 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 18013	. . 3 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑥 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)))
63	7	setc1ohomfval 49191	. . . 4 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
64	7	setc1ocofval 49192	. . . 4 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
65	7, 16, 12	funcsetc1ocl 49194	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))
66	65	func1st2nd 48936	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 1 )(2nd ‘𝐹))
67	19	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = (∅{⟨∅, ∅, 1o⟩}∅))
68	67, 54	eqtrdi 2785	. . . . 5 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = 1o)
69	14, 68	eleqtrrid 2840	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)))
70	17, 13, 37, 63, 64, 15, 66, 18, 69	isup 48964	. . 3 ⊢ (𝜑 → (𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶UP 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)⟩(𝐶UP 1 )∅)∅))
72	65	up1st2ndb 48970	. 2 ⊢ (𝜑 → (𝐼(𝐹(𝐶UP 1 )∅)∅ ↔ 𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶UP 1 )∅)∅))
73	71, 72	bitr4d 282	1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶UP 1 )∅)∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107  ∃!weu 2566  ∀wral 3050  ∃!wreu 3361  ∅c0 4313  {csn 4606  ⟨cop 4612  ⟨cotp 4614   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  1^st c1st 7994  2^nd c2nd 7995  1_oc1o 8481  Basecbs 17230  Hom chom 17285  Catccat 17679  Idccid 17680  InitOcinito 17998  SetCatcsetc 18092  Δ_funccdiag 18228  UPcup 48957  TermCatctermc 49171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-ixp 8920  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-hom 17298  df-cco 17299  df-cat 17683  df-cid 17684  df-func 17875  df-nat 17963  df-fuc 17964  df-inito 18001  df-setc 18093  df-xpc 18188  df-1stf 18189  df-curf 18230  df-diag 18232  df-up 48958  df-thinc 49119  df-termc 49172

This theorem is referenced by:  isinito2  49197