Theorem isinito2lem 49857

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 49163	. . . . 5 ⊢ (∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥) ↔ ∃!𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)⊤)
2		0ex 5254	. . . . . . . 8 ⊢ ∅ ∈ V
3		eqeq1 2741	. . . . . . . . 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 3368	. . . . . . . 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 4640	. . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ ( 1 Δfunc𝐶) = ( 1 Δfunc𝐶)
7		isinito2.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 = (SetCat‘1o)
8		setc1oterm 49850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (SetCat‘1o) ∈ TermCat
9	7, 8	eqeltri 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ TermCat
10	9	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ TermCat)
11	10	termccd 49838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ Cat)
12		isinito2lem.c	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	7	setc1obas 49851	. . . . . . . . . . . . . . . . 17 ⊢ 1o = (Base‘ 1 )
14		0lt1o 8441	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ 1o
15	14	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ 1o)
16		isinito2.f	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = ((1st ‘( 1 Δfunc𝐶))‘∅)
17		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		isinito2lem.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ (Base‘𝐶))
19	6, 11, 12, 13, 15, 16, 17, 18	diag11 18178	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1st ‘𝐹)‘𝐼) = ∅)
20	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝐼) = ∅)
21	20	opeq2d 4838	. . . . . . . . . . . . . 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 18178	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) = ∅)
27	21, 26	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥)) = (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅))
28		snex 5385	. . . . . . . . . . . . . 14 ⊢ {⟨∅, ∅, ∅⟩} ∈ V
29	28	ovsn2 49220	. . . . . . . . . . . . 13 ⊢ (⟨∅, ∅⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩}∅) = {⟨∅, ∅, ∅⟩}
30	27, 29	eqtrdi 2788	. . . . . . . . . . . 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 49838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 1 ∈ Cat)
34	12	ad2antrr 727	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → 𝐼 ∈ (Base‘𝐶))
37		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
38		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Id‘ 1 ) = (Id‘ 1 )
39		simplr 769	. . . . . . . . . . . . 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 18179	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ((Id‘ 1 )‘∅))
42	7, 38	setc1oid 49854	. . . . . . . . . . . 12 ⊢ ((Id‘ 1 )‘∅) = ∅
43	41, 42	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ((𝐼(2nd ‘𝐹)𝑥)‘𝑓) = ∅)
44		eqidd 2738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = ∅)
45	31, 43, 44	oveq123d 7389	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅) = (∅{⟨∅, ∅, ∅⟩}∅))
46	2	ovsn2 49220	. . . . . . . . . 10 ⊢ (∅{⟨∅, ∅, ∅⟩}∅) = ∅
47	45, 46	eqtr2di 2789	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) ∧ 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)) → ∅ = (((𝐼(2nd ‘𝐹)𝑥)‘𝑓)(⟨∅, ((1st ‘𝐹)‘𝐼)⟩{⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} ((1st ‘𝐹)‘𝑥))∅))
48		tbtru 1550	. . . . . . . . 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 3366	. . . . . . 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 7384	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = (∅{⟨∅, ∅, 1o⟩}∅))
53		1oex 8417	. . . . . . . . . 10 ⊢ 1o ∈ V
54	53	ovsn2 49220	. . . . . . . . 9 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = 1o
55		df1o2 8414	. . . . . . . . 9 ⊢ 1o = {∅}
56	54, 55	eqtri 2760	. . . . . . . 8 ⊢ (∅{⟨∅, ∅, 1o⟩}∅) = {∅}
57	52, 56	eqtrdi 2788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝑥)) = {∅})
58	57	raleqdv 3298	. . . . . 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 3159	. . 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 17932	. . 3 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑥 ∈ (Base‘𝐶)∃!𝑓 𝑓 ∈ (𝐼(Hom ‘𝐶)𝑥)))
63	7	setc1ohomfval 49852	. . . 4 ⊢ {⟨∅, ∅, 1o⟩} = (Hom ‘ 1 )
64	7	setc1ocofval 49853	. . . 4 ⊢ {⟨⟨∅, ∅⟩, ∅, {⟨∅, ∅, ∅⟩}⟩} = (comp‘ 1 )
65	7, 16, 12	funcsetc1ocl 49855	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 1 ))
66	65	func1st2nd 49435	. . . 4 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 1 )(2nd ‘𝐹))
67	19	oveq2d 7384	. . . . . 6 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = (∅{⟨∅, ∅, 1o⟩}∅))
68	67, 54	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)) = 1o)
69	14, 68	eleqtrrid 2844	. . . 4 ⊢ (𝜑 → ∅ ∈ (∅{⟨∅, ∅, 1o⟩} ((1st ‘𝐹)‘𝐼)))
70	17, 13, 37, 63, 64, 15, 66, 18, 69	isup 49539	. . 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 49546	. 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 1542  ⊤wtru 1543   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  ∃!wreu 3350  ∅c0 4287  {csn 4582  ⟨cop 4588  ⟨cotp 4590   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  1_oc1o 8400  Basecbs 17148  Hom chom 17200  Catccat 17599  Idccid 17600  InitOcinito 17917  SetCatcsetc 18011  Δ_funccdiag 18147   UP cup 49532  TermCatctermc 49831

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-hom 17213  df-cco 17214  df-cat 17603  df-cid 17604  df-func 17794  df-nat 17882  df-fuc 17883  df-inito 17920  df-setc 18012  df-xpc 18107  df-1stf 18108  df-curf 18149  df-diag 18151  df-up 49533  df-thinc 49777  df-termc 49832

This theorem is referenced by:  isinito2  49858