Theorem isinito2lem 49988

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 49294	. . . . 5 ⊢ (∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤)
2		0ex 5229	. . . . . . . 8 ⊢ ∅ ∈ V
3		eqeq1 2743	. . . . . . . . 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 3360	. . . . . . . 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 4613	. . . . . . 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 2739	. . . . . . . . . . . . . . . . 17 ⊢ ( 1 Δfunc𝐶) = ( 1 Δfunc𝐶)
7		isinito2.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 = (SetCat‘1o)
8		setc1oterm 49981	. . . . . . . . . . . . . . . . . . . 20 ⊢ (SetCat‘1o) ∈ TermCat
9	7, 8	eqeltri 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ TermCat
10	9	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ TermCat)
11	10	termccd 49969	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ Cat)
12		isinito2lem.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	7	setc1obas 49982	. . . . . . . . . . . . . . . . 17 ⊢ 1o = (Base‘ 1 )
14		0lt1o 8429	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 1o
15	14	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 1o)
16		isinito2.f	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)
17		eqid 2739	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		isinito2lem.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ (Base‘𝐶))
19	6, 11, 12, 13, 15, 16, 17, 18	diag11 18200	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1st ‘𝐹)‘𝐼) = ∅)
20	19	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝐼) = ∅)
21	20	opeq2d 4811	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨∅, ((1st ‘𝐹)‘𝐼)⟩ = ⟨∅, ∅⟩)
22	11	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 1 ∈ Cat)
23	12	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
24	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ∅ ∈ 1o)
25		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
26	6, 22, 23, 13, 24, 16, 17, 25	diag11 18200	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) = ∅)
27	21, 26	oveq12d 7374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
28		snex 5368	. . . . . . . . . . . . . 14 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
29	28	ovsn2 49351	. . . . . . . . . . . . 13 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
30	27, 29	eqtrdi 2790	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = {⟨∅, ∅, ∅⟩})
31	30	adantr 481	. . . . . . . . . . 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 49969	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 1 ∈ Cat)
34	12	ad2antrr 732	. . . . . . . . . . . . 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 732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝐼 ∈ (Base‘𝐶))
37		eqid 2739	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38		eqid 2739	. . . . . . . . . . . . 13 ⊢ (Id‘ 1 ) = (Id‘ 1 )
39		simplr 774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
40		simpr 485	. . . . . . . . . . . . 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 18201	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ((Id‘ 1 )‘∅))
42	7, 38	setc1oid 49985	. . . . . . . . . . . 12 ⊢ ((Id‘ 1 )‘∅) = ∅
43	41, 42	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ∅)
44		eqidd 2740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = ∅)
45	31, 43, 44	oveq123d 7377	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) = (∅{⟨∅, ∅, ∅⟩}∅))
46	2	ovsn2 49351	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
47	45, 46	eqtr2di 2791	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅))
48		tbtru 1555	. . . . . . . . 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 219	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) ↔ ⊤))
50	49	reubidva 3358	. . . . . . 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 285	. . . . . 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 7372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = (∅{⟨∅, ∅, 1o⟩}∅))
53		1oex 8405	. . . . . . . . . 10 ⊢ 1o ∈ V
54	53	ovsn2 49351	. . . . . . . . 9 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
55		df1o2 8402	. . . . . . . . 9 ⊢ 1o = {∅}
56	54, 55	eqtri 2762	. . . . . . . 8 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = {∅}
57	52, 56	eqtrdi 2790	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = {∅})
58	57	raleqdv 3297	. . . . . 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 283	. . . . 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 284	. . . 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 3160	. . 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 17954	. . 3 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑥 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)))
63	7	setc1ohomfval 49983	. . . 4 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
64	7	setc1ocofval 49984	. . . 4 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
65	7, 16, 12	funcsetc1ocl 49986	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))
66	65	func1st2nd 49566	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 1 )(2nd ‘𝐹))
67	19	oveq2d 7372	. . . . . 6 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = (∅{⟨∅, ∅, 1o⟩}∅))
68	67, 54	eqtrdi 2790	. . . . 5 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = 1o)
69	14, 68	eleqtrrid 2846	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)))
70	17, 13, 37, 63, 64, 15, 66, 18, 69	isup 49670	. . 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 312	. 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 1 )∅)∅))
72	65	up1st2ndb 49677	. 2 ⊢ (𝜑 → (𝐼(𝐹(𝐶 UP 1 )∅)∅ ↔ 𝐼(⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 UP 1 )∅)∅))
73	71, 72	bitr4d 283	1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼(𝐹(𝐶 UP 1 )∅)∅))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  ∃!weu 2572  ∀wral 3053  ∃!wreu 3342  ∅c0 4261  {csn 4555  ⟨cop 4561  ⟨cotp 4563   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  1_oc1o 8388  Basecbs 17170  Hom chom 17222  Catccat 17621  Idccid 17622  InitOcinito 17939  SetCatcsetc 18033  Δ_funccdiag 18169   UP cup 49663  TermCatctermc 49962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-func 17816  df-nat 17904  df-fuc 17905  df-inito 17942  df-setc 18034  df-xpc 18129  df-1stf 18130  df-curf 18171  df-diag 18173  df-up 49664  df-thinc 49908  df-termc 49963

This theorem is referenced by:  isinito2  49989