fuciso - Metamath Proof Explorer

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Theorem fuciso 18092

Description: A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (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))
fuciso.i	⊢ 𝐼 = (Iso‘𝑄)
fuciso.j	⊢ 𝐽 = (Iso‘𝐷)

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

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

Proof of Theorem fuciso Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuciso.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2	1	fucbas 18077	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 18078	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
5		fuciso.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝑄)
6		fuciso.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 17977	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9	8	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
10	8	simprd 501	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
11	1, 9, 10	fuccat 18087	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
12		fuciso.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
13	2, 4, 5, 11, 6, 12	isohom 17890	. . . 4 ⊢ (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
14	13	sselda 3931	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15		eqid 2760	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2760	. . . . 5 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17	10	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
18		fuciso.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
19		relfunc 17976	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
20		1st2ndbr 8044	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
21	19, 6, 20	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
22	18, 15, 21	funcf1 17980	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
23	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
24	23	ffvelcdmda 7080	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
25		1st2ndbr 8044	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
26	19, 12, 25	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
27	18, 15, 26	funcf1 17980	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
28	27	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
29	28	ffvelcdmda 7080	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30		fuciso.j	. . . . 5 ⊢ 𝐽 = (Iso‘𝐷)
31		eqid 2760	. . . . . . . . . . . 12 ⊢ (Inv‘𝑄) = (Inv‘𝑄)
32	2, 31, 11, 6, 12, 5	isoval 17879	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
33	32	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
34	2, 31, 11, 6, 12	invfun 17878	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35		funfvbrb 7046	. . . . . . . . . . 11 ⊢ (Fun (𝐹(Inv‘𝑄)𝐺) → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
36	34, 35	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
37	33, 36	bitrd 282	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
38	37	biimpa 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴))
39	1, 18, 3, 6, 12, 31, 16	fucinv 18090	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥)(((1^st ‘𝐹)‘𝑥)(Inv‘𝐷)((1^st ‘𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))))
40	39	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥)(((1^st ‘𝐹)‘𝑥)(Inv‘𝐷)((1^st ‘𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))))
41	38, 40	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥)(((1^st ‘𝐹)‘𝑥)(Inv‘𝐷)((1^st ‘𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥)))
42	41	simp3d 1162	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥)(((1^st ‘𝐹)‘𝑥)(Inv‘𝐷)((1^st ‘𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
43	42	r19.21bi 3254	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥)(((1^st ‘𝐹)‘𝑥)(Inv‘𝐷)((1^st ‘𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
44	15, 16, 17, 24, 29, 30, 43	inviso1 17880	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
45	44	ralrimiva 3154	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
46	14, 45	jca 521	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))))
47	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝑄 ∈ Cat)
48	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐹 ∈ (𝐶 Func 𝐷))
49	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐺 ∈ (𝐶 Func 𝐷))
50		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
51	10	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
52	22	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
53	52	ffvelcdmda 7080	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
54	27	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
55	54	ffvelcdmda 7080	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
56		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
57		fveq2 6881	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
58		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
59		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
60	58, 59	oveq12d 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) = (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
61	57, 60	eleq12d 2854	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ↔ (𝐴‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦))))
62	61	rspccva 3575	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ 𝑦 ∈ 𝐵) → (𝐴‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
63	56, 62	sylan 592	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → (𝐴‘𝑦) ∈ (((1^st ‘𝐹)‘𝑦)𝐽((1^st ‘𝐺)‘𝑦)))
64	15, 30, 16, 51, 53, 55, 63	invisoinvr 17905	. . . 4 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → (𝐴‘𝑦)(((1^st ‘𝐹)‘𝑦)(Inv‘𝐷)((1^st ‘𝐺)‘𝑦))((((1^st ‘𝐹)‘𝑦)(Inv‘𝐷)((1^st ‘𝐺)‘𝑦))‘(𝐴‘𝑦)))
65	1, 18, 3, 48, 49, 31, 16, 50, 64	invfuc 18091	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴(𝐹(Inv‘𝑄)𝐺)(𝑦 ∈ 𝐵 ↦ ((((1^st ‘𝐹)‘𝑦)(Inv‘𝐷)((1^st ‘𝐺)‘𝑦))‘(𝐴‘𝑦))))
66	2, 31, 47, 48, 49, 5, 65	inviso1 17880	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
67	46, 66	impbida 813	1 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 Rel wrel 5660 Fun wfun 6529 ⟶wf 6531 ‘cfv 6535 (class class class)co 7416 1^st c1st 7990 2^nd c2nd 7991 Basecbs 17326 Catccat 17777 Invcinv 17859 Isociso 17860 Func cfunc 17968 Nat cnat 18058 FuncCat cfuc 18059

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-map 8835 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-fz 13587 df-struct 17264 df-slot 17299 df-ndx 17311 df-base 17327 df-hom 17391 df-cco 17392 df-cat 17781 df-cid 17782 df-sect 17861 df-inv 17862 df-iso 17863 df-func 17972 df-nat 18060 df-fuc 18061

This theorem is used by: yonffthlem 18395

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