fucinv - Metamath Proof Explorer

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

Theorem fucinv 17930

Description: Two natural transformations are inverses of each other iff all the components are inverse. (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))
fucinv.i	⊢ 𝐼 = (Inv‘𝑄)
fucinv.j	⊢ 𝐽 = (Inv‘𝐷)

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

Distinct variable groups: 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐼 𝑥,𝐹 𝑥,𝐺 𝑥,𝐽 𝑥,𝑁 𝑥,𝑉 𝜑,𝑥 𝑥,𝑄 𝑥,𝑈

**Proof of Theorem fucinv**
Step	Hyp	Ref	Expression
1		fuciso.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		fuciso.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		fuciso.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		fuciso.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5		fuciso.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
6		eqid 2724	. . . 4 ⊢ (Sect‘𝑄) = (Sect‘𝑄)
7		eqid 2724	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
8	1, 2, 3, 4, 5, 6, 7	fucsect 17929	. . 3 ⊢ (𝜑 → (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥))))
9	1, 2, 3, 5, 4, 6, 7	fucsect 17929	. . 3 ⊢ (𝜑 → (𝑉(𝐺(Sect‘𝑄)𝐹)𝑈 ↔ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
10	8, 9	anbi12d 630	. 2 ⊢ (𝜑 → ((𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ∧ 𝑉(𝐺(Sect‘𝑄)𝐹)𝑈) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))))
11	1	fucbas 17916	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
12		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
13		funcrcl 17814	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	4, 13	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
15	14	simpld 494	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	14	simprd 495	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	1, 15, 16	fuccat 17927	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
18	11, 12, 17, 4, 5, 6	isinv 17708	. 2 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ∧ 𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19		eqid 2724	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		fucinv.j	. . . . . . 7 ⊢ 𝐽 = (Inv‘𝐷)
21	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
22		relfunc 17813	. . . . . . . . . 10 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 8022	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
24	22, 4, 23	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
25	2, 19, 24	funcf1 17817	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
26	25	ffvelcdmda 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
27		1st2ndbr 8022	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
28	22, 5, 27	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
29	2, 19, 28	funcf1 17817	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
30	29	ffvelcdmda 7077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
31	19, 20, 21, 26, 30, 7	isinv 17708	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ↔ ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
32	31	ralbidva 3167	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
33		r19.26 3103	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ((𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ↔ (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))
34	32, 33	bitrdi 287	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ↔ (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
35	34	anbi2d 628	. . 3 ⊢ (𝜑 → (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))))
36		df-3an 1086	. . 3 ⊢ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)))
37		df-3an 1086	. . . . 5 ⊢ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)))
38		3ancoma 1095	. . . . . 6 ⊢ ((𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))
39		df-3an 1086	. . . . . 6 ⊢ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))
40	38, 39	bitri 275	. . . . 5 ⊢ ((𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))
41	37, 40	anbi12i 626	. . . 4 ⊢ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))) ↔ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ∧ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
42		anandi 673	. . . 4 ⊢ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))) ↔ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ∧ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
43	41, 42	bitr4i 278	. . 3 ⊢ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥))))
44	35, 36, 43	3bitr4g 314	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)(Sect‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝑉‘𝑥)(((1^st ‘𝐺)‘𝑥)(Sect‘𝐷)((1^st ‘𝐹)‘𝑥))(𝑈‘𝑥)))))
45	10, 18, 44	3bitr4d 311	1 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝑈‘𝑥)(((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))(𝑉‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3053 class class class wbr 5139 Rel wrel 5672 ‘cfv 6534 (class class class)co 7402 1^st c1st 7967 2^nd c2nd 7968 Basecbs 17145 Catccat 17609 Sectcsect 17692 Invcinv 17693 Func cfunc 17805 Nat cnat 17896 FuncCat cfuc 17897

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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-struct 17081 df-slot 17116 df-ndx 17128 df-base 17146 df-hom 17222 df-cco 17223 df-cat 17613 df-cid 17614 df-sect 17695 df-inv 17696 df-func 17809 df-nat 17898 df-fuc 17899

This theorem is referenced by: invfuc 17931 fuciso 17932

W3C validator