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Theorem fucinv 17883

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 2729	. . . 4 ⊢ (Sect‘𝑄) = (Sect‘𝑄)
7		eqid 2729	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
8	1, 2, 3, 4, 5, 6, 7	fucsect 17882	. . 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 17882	. . 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 17870	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
12		fucinv.i	. . 3 ⊢ 𝐼 = (Inv‘𝑄)
13		funcrcl 17770	. . . . . 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 17880	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
18	11, 12, 17, 4, 5, 6	isinv 17667	. 2 ⊢ (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ∧ 𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
20		fucinv.j	. . . . . . 7 ⊢ 𝐽 = (Inv‘𝐷)
21	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
22		relfunc 17769	. . . . . . . . . 10 ⊢ Rel (𝐶 Func 𝐷)
23		1st2ndbr 7977	. . . . . . . . . 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 17773	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
26	25	ffvelcdmda 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
27		1st2ndbr 7977	. . . . . . . . . 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 17773	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
30	29	ffvelcdmda 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
31	19, 20, 21, 26, 30, 7	isinv 17667	. . . . . 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 3150	. . . . 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 3089	. . . . 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 3044 class class class wbr 5092 Rel wrel 5624 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 Basecbs 17120 Catccat 17570 Sectcsect 17651 Invcinv 17652 Func cfunc 17761 Nat cnat 17851 FuncCat cfuc 17852

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-cid 17575 df-sect 17654 df-inv 17655 df-func 17765 df-nat 17853 df-fuc 17854

This theorem is referenced by: invfuc 17884 fuciso 17885

W3C validator