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Theorem fucinv 17994

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 2736	. . . 4 ⊢ (Sect‘𝑄) = (Sect‘𝑄)
7		eqid 2736	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
8	1, 2, 3, 4, 5, 6, 7	fucsect 17993	. . 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 17993	. . 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 632	. 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 17981	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
12		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
13		funcrcl 17881	. . . . . 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 17991	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
18	11, 12, 17, 4, 5, 6	isinv 17778	. 2 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ∧ 𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19		eqid 2736	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		fucinv.j	. . . . . . 7 ⊢ 𝐽 = (Inv‘𝐷)
21	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
22		relfunc 17880	. . . . . . . . . 10 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 8046	. . . . . . . . . 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 17884	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
26	25	ffvelcdmda 7079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
27		1st2ndbr 8046	. . . . . . . . . 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 17884	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
30	29	ffvelcdmda 7079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
31	19, 20, 21, 26, 30, 7	isinv 17778	. . . . . 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 3162	. . . . 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 3099	. . . . 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 630	. . 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 1088	. . 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 1088	. . . . 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 1097	. . . . . 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 1088	. . . . . 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 628	. . . 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 676	. . . 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 class class class wbr 5124 Rel wrel 5664 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 Basecbs 17233 Catccat 17681 Sectcsect 17762 Invcinv 17763 Func cfunc 17872 Nat cnat 17962 FuncCat cfuc 17963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-hom 17300 df-cco 17301 df-cat 17685 df-cid 17686 df-sect 17765 df-inv 17766 df-func 17876 df-nat 17964 df-fuc 17965

This theorem is referenced by: invfuc 17995 fuciso 17996

W3C validator