fuciso - Metamath Proof Explorer

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Theorem fuciso 18179

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 18164	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 18165	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
5		fuciso.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝑄)
6		fuciso.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 18064	. . . . . . . 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 18174	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
12		fuciso.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
13	2, 4, 5, 11, 6, 12	isohom 17977	. . . 4 ⊢ (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
14	13	sselda 3913	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15		eqid 2763	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2763	. . . . 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 18063	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
20		1st2ndbr 8074	. . . . . . . . 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 18067	. . . . . . 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 7105	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
25		1st2ndbr 8074	. . . . . . . . 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 18067	. . . . . . 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 7105	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30		fuciso.j	. . . . 5 ⊢ 𝐽 = (Iso‘𝐷)
31		eqid 2763	. . . . . . . . . . . 12 ⊢ (Inv‘𝑄) = (Inv‘𝑄)
32	2, 31, 11, 6, 12, 5	isoval 17966	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
33	32	eleq2d 2850	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
34	2, 31, 11, 6, 12	invfun 17965	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35		funfvbrb 7070	. . . . . . . . . . 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 18177	. . . . . . . . 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 3258	. . . . 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 17967	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
45	44	ralrimiva 3158	. . 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 7105	. . . . 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 7105	. . . . 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 6905	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
58		fveq2 6905	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
59		fveq2 6905	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
60	58, 59	oveq12d 7458	. . . . . . . 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 2858	. . . . . . 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 3579	. . . . . 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 17992	. . . 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 18178	. . 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 17967	. 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 2147 ∀wral 3080 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 Rel wrel 5660 Fun wfun 6549 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 1^st c1st 8020 2^nd c2nd 8021 Basecbs 17413 Catccat 17864 Invcinv 17946 Isociso 17947 Func cfunc 18055 Nat cnat 18145 FuncCat cfuc 18146

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 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 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-er 8742 df-map 8874 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-fz 13664 df-struct 17351 df-slot 17386 df-ndx 17398 df-base 17414 df-hom 17478 df-cco 17479 df-cat 17868 df-cid 17869 df-sect 17948 df-inv 17949 df-iso 17950 df-func 18059 df-nat 18147 df-fuc 18148

This theorem is used by: yonffthlem 18482

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