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Theorem fucsect 17955

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 17942	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 17943	. . 3 ⊢ 𝑁 = (Hom ‘𝑄)
5		eqid 2727	. . 3 ⊢ (comp‘𝑄) = (comp‘𝑄)
6		eqid 2727	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
7		fucsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝑄)
8		fuciso.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17840	. . . . . 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 17953	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
14		fuciso.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
15	2, 4, 5, 6, 7, 13, 8, 14	issect 17727	. 2 ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16		ovex 7447	. . . . . . 7 ⊢ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
17	16	rgenw 3060	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
18		mpteqb 7018	. . . . . 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 2727	. . . . . . 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 17945	. . . . . 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 2727	. . . . . . . 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 17954	. . . . . . 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 2727	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
30	29, 25	cidfn 17650	. . . . . . . . . 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 6718	. . . . . . . . 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 17839	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
35		1st2ndbr 8040	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
36	34, 8, 35	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
37	20, 29, 36	funcf1 17843	. . . . . . . . 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 7136	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝐷) ∘ (1^st ‘𝐹)) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
41	27, 40	eqtrd 2767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
42	24, 41	eqeq12d 2743	. . . . 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 2727	. . . . . . 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 7088	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
47		1st2ndbr 8040	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
48	34, 14, 47	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
49	20, 29, 48	funcf1 17843	. . . . . . . . 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 7088	. . . . . . 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 17932	. . . . . . . 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 17934	. . . . . . 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 17932	. . . . . . . 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 17934	. . . . . . 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 17728	. . . . . 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 3170	. . . . 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 578	. . 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 1087	. . 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 1087	. . 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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ⟨cop 4630 class class class wbr 5142 ↦ cmpt 5225 ∘ ccom 5676 Rel wrel 5677 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17171 Hom chom 17235 compcco 17236 Catccat 17635 Idccid 17636 Sectcsect 17718 Func cfunc 17831 Nat cnat 17922 FuncCat cfuc 17923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-hom 17248 df-cco 17249 df-cat 17639 df-cid 17640 df-sect 17721 df-func 17835 df-nat 17924 df-fuc 17925

This theorem is referenced by: fucinv 17956

W3C validator