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Theorem fucinv 17876

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 2731	. . . 4 ⊢ (Sect‘𝑄) = (Sect‘𝑄)
7		eqid 2731	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
8	1, 2, 3, 4, 5, 6, 7	fucsect 17875	. . 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 17875	. . 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 631	. 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 17862	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
12		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
13		funcrcl 17763	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
14	4, 13	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
15	14	simpld 495	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	14	simprd 496	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	1, 15, 16	fuccat 17873	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
18	11, 12, 17, 4, 5, 6	isinv 17657	. 2 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ∧ 𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		fucinv.j	. . . . . . 7 ⊢ 𝐽 = (Inv‘𝐷)
21	16	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
22		relfunc 17762	. . . . . . . . . 10 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 7979	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
24	22, 4, 23	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
25	2, 19, 24	funcf1 17766	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
26	25	ffvelcdmda 7040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
27		1st2ndbr 7979	. . . . . . . . . 10 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
28	22, 5, 27	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
29	2, 19, 28	funcf1 17766	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
30	29	ffvelcdmda 7040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
31	19, 20, 21, 26, 30, 7	isinv 17657	. . . . . 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 3168	. . . . 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 3110	. . . . 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 629	. . 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 1089	. . 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 1089	. . . . 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 1098	. . . . . 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 1089	. . . . . 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 627	. . . 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 674	. . . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 class class class wbr 5110 Rel wrel 5643 ‘cfv 6501 (class class class)co 7362 1^st c1st 7924 2^nd c2nd 7925 Basecbs 17094 Catccat 17558 Sectcsect 17641 Invcinv 17642 Func cfunc 17754 Nat cnat 17842 FuncCat cfuc 17843

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 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 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-fz 13435 df-struct 17030 df-slot 17065 df-ndx 17077 df-base 17095 df-hom 17171 df-cco 17172 df-cat 17562 df-cid 17563 df-sect 17644 df-inv 17645 df-func 17758 df-nat 17844 df-fuc 17845

This theorem is referenced by: invfuc 17877 fuciso 17878

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