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Theorem fucsect 17971

Description: Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)

**Hypotheses**
Ref	Expression
fuciso.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fuciso.b	⊢ 𝐵 = (Base‘𝐶)
fuciso.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fuciso.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
fucsect.s	⊢ 𝑆 = (Sect‘𝑄)
fucsect.t	⊢ 𝑇 = (Sect‘𝐷)

**Assertion**
Ref	Expression
fucsect	⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥))))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐹 𝑥,𝐺 𝑥,𝑁 𝑥,𝑉 𝜑,𝑥 𝑥,𝑄 𝑥,𝑈 Allowed substitution hints: 𝑆(𝑥) 𝑇(𝑥)

**Proof of Theorem fucsect**
Step	Hyp	Ref	Expression
1		fuciso.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2	1	fucbas 17958	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 17959	. . 3 ⊢ 𝑁 = (Hom ‘𝑄)
5		eqid 2728	. . 3 ⊢ (comp‘𝑄) = (comp‘𝑄)
6		eqid 2728	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
7		fucsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝑄)
8		fuciso.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17856	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	10	simpld 493	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	simprd 494	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	1, 11, 12	fuccat 17969	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
14		fuciso.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
15	2, 4, 5, 6, 7, 13, 8, 14	issect 17743	. 2 ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16		ovex 7459	. . . . . . 7 ⊢ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
17	16	rgenw 3062	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
18		mpteqb 7029	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V → ((𝑥 ∈ 𝐵 ↦ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
19	17, 18	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((𝑥 ∈ 𝐵 ↦ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
20		fuciso.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
21		eqid 2728	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
22		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑈 ∈ (𝐹𝑁𝐺))
23		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑉 ∈ (𝐺𝑁𝐹))
24	1, 3, 20, 21, 5, 22, 23	fucco 17961	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = (𝑥 ∈ 𝐵 ↦ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
25		eqid 2728	. . . . . . . 8 ⊢ (Id‘𝐷) = (Id‘𝐷)
26	8	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐹 ∈ (𝐶 Func 𝐷))
27	1, 6, 25, 26	fucid 17970	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
28	12	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐷 ∈ Cat)
29		eqid 2728	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
30	29, 25	cidfn 17666	. . . . . . . . . 10 ⊢ (𝐷 ∈ Cat → (Id‘𝐷) Fn (Base‘𝐷))
31	28, 30	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷) Fn (Base‘𝐷))
32		dffn2 6729	. . . . . . . . 9 ⊢ ((Id‘𝐷) Fn (Base‘𝐷) ↔ (Id‘𝐷):(Base‘𝐷)⟶V)
33	31, 32	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷):(Base‘𝐷)⟶V)
34		relfunc 17855	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
35		1st2ndbr 8052	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
36	34, 8, 35	sylancr 585	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
37	20, 29, 36	funcf1 17859	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
38	37	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
39		fcompt 7148	. . . . . . . 8 ⊢ (((Id‘𝐷):(Base‘𝐷)⟶V ∧ (1^st ‘𝐹):𝐵⟶(Base‘𝐷)) → ((Id‘𝐷) ∘ (1^st ‘𝐹)) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
40	33, 38, 39	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝐷) ∘ (1^st ‘𝐹)) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
41	27, 40	eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
42	24, 41	eqeq12d 2744	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹) ↔ (𝑥 ∈ 𝐵 ↦ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥)))))
43		eqid 2728	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
44		fucsect.t	. . . . . . 7 ⊢ 𝑇 = (Sect‘𝐷)
45	28	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
46	38	ffvelcdmda 7099	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
47		1st2ndbr 8052	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
48	34, 14, 47	sylancr 585	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
49	20, 29, 48	funcf1 17859	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
50	49	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
51	50	ffvelcdmda 7099	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
52	22	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ (𝐹𝑁𝐺))
53	3, 52	nat1st2nd 17948	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
54		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
55	3, 53, 20, 43, 54	natcl 17950	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
56	23	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑉 ∈ (𝐺𝑁𝐹))
57	3, 56	nat1st2nd 17948	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑉 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
58	3, 57, 20, 43, 54	natcl 17950	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → (𝑉‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑥)))
59	29, 43, 21, 25, 44, 45, 46, 51, 55, 58	issect2 17744	. . . . . 6 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ↔ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
60	59	ralbidva 3173	. . . . 5 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
61	19, 42, 60	3bitr4d 310	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹) ↔ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)))
62	61	pm5.32da 577	. . 3 ⊢ (𝜑 → (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥))))
63		df-3an 1086	. . 3 ⊢ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)))
64		df-3an 1086	. . 3 ⊢ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)))
65	62, 63, 64	3bitr4g 313	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹)) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥))))
66	15, 65	bitrd 278	1 ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝑇((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ⟨cop 4638 class class class wbr 5152 ↦ cmpt 5235 ∘ ccom 5686 Rel wrel 5687 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 1^st c1st 7997 2^nd c2nd 7998 Basecbs 17187 Hom chom 17251 compcco 17252 Catccat 17651 Idccid 17652 Sectcsect 17734 Func cfunc 17847 Nat cnat 17938 FuncCat cfuc 17939

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-hom 17264 df-cco 17265 df-cat 17655 df-cid 17656 df-sect 17737 df-func 17851 df-nat 17940 df-fuc 17941

This theorem is referenced by: fucinv 17972

W3C validator