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Theorem fucsect 17937

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 17925	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 17926	. . 3 ⊢ 𝑁 = (Hom ‘𝑄)
5		eqid 2729	. . 3 ⊢ (comp‘𝑄) = (comp‘𝑄)
6		eqid 2729	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
7		fucsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝑄)
8		fuciso.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17825	. . . . . 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 17935	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
14		fuciso.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
15	2, 4, 5, 6, 7, 13, 8, 14	issect 17715	. 2 ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16		ovex 7420	. . . . . . 7 ⊢ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
17	16	rgenw 3048	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
18		mpteqb 6987	. . . . . 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 2729	. . . . . . 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 17927	. . . . . 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 2729	. . . . . . . 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 17936	. . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
30	29, 25	cidfn 17640	. . . . . . . . . 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 6690	. . . . . . . . 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 17824	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
35		1st2ndbr 8021	. . . . . . . . . . 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 17828	. . . . . . . . 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 7105	. . . . . . . 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 2764	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
42	24, 41	eqeq12d 2745	. . . . 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 2729	. . . . . . 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 7056	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
47		1st2ndbr 8021	. . . . . . . . . . 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 17828	. . . . . . . . 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 7056	. . . . . . 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 17916	. . . . . . . 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 17918	. . . . . . 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 17916	. . . . . . . 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 17918	. . . . . . 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 17716	. . . . . 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 3154	. . . . 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 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 ⟨cop 4595 class class class wbr 5107 ↦ cmpt 5188 ∘ ccom 5642 Rel wrel 5643 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 Basecbs 17179 Hom chom 17231 compcco 17232 Catccat 17625 Idccid 17626 Sectcsect 17706 Func cfunc 17816 Nat cnat 17906 FuncCat cfuc 17907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17629 df-cid 17630 df-sect 17709 df-func 17820 df-nat 17908 df-fuc 17909

This theorem is referenced by: fucinv 17938

W3C validator