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Theorem fucsect 17879

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 17867	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 17868	. . 3 ⊢ 𝑁 = (Hom ‘𝑄)
5		eqid 2731	. . 3 ⊢ (comp‘𝑄) = (comp‘𝑄)
6		eqid 2731	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
7		fucsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝑄)
8		fuciso.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17767	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	10	simpld 494	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	simprd 495	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	1, 11, 12	fuccat 17877	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
14		fuciso.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
15	2, 4, 5, 6, 7, 13, 8, 14	issect 17657	. 2 ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16		ovex 7379	. . . . . . 7 ⊢ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
17	16	rgenw 3051	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
18		mpteqb 6948	. . . . . 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 2731	. . . . . . 7 ⊢ (comp‘𝐷) = (comp‘𝐷)
22		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑈 ∈ (𝐹𝑁𝐺))
23		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝑉 ∈ (𝐺𝑁𝐹))
24	1, 3, 20, 21, 5, 22, 23	fucco 17869	. . . . . 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 2731	. . . . . . . 8 ⊢ (Id‘𝐷) = (Id‘𝐷)
26	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐹 ∈ (𝐶 Func 𝐷))
27	1, 6, 25, 26	fucid 17878	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
28	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → 𝐷 ∈ Cat)
29		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
30	29, 25	cidfn 17582	. . . . . . . . . 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 6653	. . . . . . . . 9 ⊢ ((Id‘𝐷) Fn (Base‘𝐷) ↔ (Id‘𝐷):(Base‘𝐷)⟶V)
33	31, 32	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (Id‘𝐷):(Base‘𝐷)⟶V)
34		relfunc 17766	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
35		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
36	34, 8, 35	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
37	20, 29, 36	funcf1 17770	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
38	37	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
39		fcompt 7066	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝐷) ∘ (1^st ‘𝐹)) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
41	27, 40	eqtrd 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
42	24, 41	eqeq12d 2747	. . . . 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 2731	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
44		fucsect.t	. . . . . . 7 ⊢ 𝑇 = (Sect‘𝐷)
45	28	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
46	38	ffvelcdmda 7017	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
47		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
48	34, 14, 47	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
49	20, 29, 48	funcf1 17770	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
50	49	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
51	50	ffvelcdmda 7017	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
52	22	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ (𝐹𝑁𝐺))
53	3, 52	nat1st2nd 17858	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
54		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
55	3, 53, 20, 43, 54	natcl 17860	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → (𝑈‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
56	23	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → 𝑉 ∈ (𝐺𝑁𝐹))
57	3, 56	nat1st2nd 17858	. . . . . . . 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 17860	. . . . . . 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 17658	. . . . . 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 3153	. . . . 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 311	. . . 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 579	. . 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 1088	. . 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 1088	. . 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 314	. 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 279	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ⟨cop 4582 class class class wbr 5091 ↦ cmpt 5172 ∘ ccom 5620 Rel wrel 5621 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 Basecbs 17117 Hom chom 17169 compcco 17170 Catccat 17567 Idccid 17568 Sectcsect 17648 Func cfunc 17758 Nat cnat 17848 FuncCat cfuc 17849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-fz 13405 df-struct 17055 df-slot 17090 df-ndx 17102 df-base 17118 df-hom 17182 df-cco 17183 df-cat 17571 df-cid 17572 df-sect 17651 df-func 17762 df-nat 17850 df-fuc 17851

This theorem is referenced by: fucinv 17880

W3C validator