fucsect - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fucsect		Structured version Visualization version GIF version

Theorem fucsect 18042

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 18029	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 18030	. . 3 ⊢ 𝑁 = (Hom ‘𝑄)
5		eqid 2740	. . 3 ⊢ (comp‘𝑄) = (comp‘𝑄)
6		eqid 2740	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
7		fucsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝑄)
8		fuciso.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17927	. . . . . 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 18040	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
14		fuciso.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
15	2, 4, 5, 6, 7, 13, 8, 14	issect 17814	. 2 ⊢ (𝜑 → (𝑈(𝐹𝑆𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ (𝑉(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐹)𝑈) = ((Id‘𝑄)‘𝐹))))
16		ovex 7481	. . . . . . 7 ⊢ ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
17	16	rgenw 3071	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ((𝑉‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ∈ V
18		mpteqb 7048	. . . . . 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 2740	. . . . . . 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 18032	. . . . . 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 2740	. . . . . . . 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 18041	. . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
30	29, 25	cidfn 17737	. . . . . . . . . 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 6749	. . . . . . . . 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 17926	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
35		1st2ndbr 8083	. . . . . . . . . . 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 17930	. . . . . . . . 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 7167	. . . . . . . 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 2780	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) → ((Id‘𝑄)‘𝐹) = (𝑥 ∈ 𝐵 ↦ ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
42	24, 41	eqeq12d 2756	. . . . 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 2740	. . . . . . 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 7118	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹))) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
47		1st2ndbr 8083	. . . . . . . . . . 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 17930	. . . . . . . . 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 7118	. . . . . . 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 18019	. . . . . . . 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 18021	. . . . . . 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 18019	. . . . . . . 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 18021	. . . . . . 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 17815	. . . . . 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 3182	. . . . 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 1089	. . 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 1089	. . 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 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⟨cop 4654 class class class wbr 5166 ↦ cmpt 5249 ∘ ccom 5704 Rel wrel 5705 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 Basecbs 17258 Hom chom 17322 compcco 17323 Catccat 17722 Idccid 17723 Sectcsect 17805 Func cfunc 17918 Nat cnat 18009 FuncCat cfuc 18010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-hom 17335 df-cco 17336 df-cat 17726 df-cid 17727 df-sect 17808 df-func 17922 df-nat 18011 df-fuc 18012

This theorem is referenced by: fucinv 18043

W3C validator