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Theorem fucinv 17607

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 2738	. . . 4 ⊢ (Sect‘𝑄) = (Sect‘𝑄)
7		eqid 2738	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
8	1, 2, 3, 4, 5, 6, 7	fucsect 17606	. . 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 17606	. . 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 17593	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
12		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
13		funcrcl 17494	. . . . . 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 17604	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
18	11, 12, 17, 4, 5, 6	isinv 17389	. 2 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ∧ 𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		fucinv.j	. . . . . . 7 ⊢ 𝐽 = (Inv‘𝐷)
21	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
22		relfunc 17493	. . . . . . . . . 10 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 7856	. . . . . . . . . 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 17497	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
26	25	ffvelrnda 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
27		1st2ndbr 7856	. . . . . . . . . 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 17497	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
30	29	ffvelrnda 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
31	19, 20, 21, 26, 30, 7	isinv 17389	. . . . . 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 3119	. . . . 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 3094	. . . . 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 286	. . . 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 1087	. . 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 1087	. . . . 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 1096	. . . . . 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 1087	. . . . . 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 274	. . . . 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 672	. . . 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 277	. . 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 313	. 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 310	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 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 class class class wbr 5070 Rel wrel 5585 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 Catccat 17290 Sectcsect 17373 Invcinv 17374 Func cfunc 17485 Nat cnat 17573 FuncCat cfuc 17574

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-hom 16912 df-cco 16913 df-cat 17294 df-cid 17295 df-sect 17376 df-inv 17377 df-func 17489 df-nat 17575 df-fuc 17576

This theorem is referenced by: invfuc 17608 fuciso 17609

W3C validator