fuciso - Metamath Proof Explorer

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Theorem fuciso 18035

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 18020	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 18021	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
5		fuciso.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝑄)
6		fuciso.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 17920	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9	8	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
10	8	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
11	1, 9, 10	fuccat 18030	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
12		fuciso.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
13	2, 4, 5, 11, 6, 12	isohom 17833	. . . 4 ⊢ (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
14	13	sselda 3945	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15		eqid 2769	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2769	. . . . 5 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17	10	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
18		fuciso.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
19		relfunc 17919	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
20		1st2ndbr 8039	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
21	19, 6, 20	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
22	18, 15, 21	funcf1 17923	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
23	22	adantr 485	. . . . . 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 8039	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
26	19, 12, 25	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
27	18, 15, 26	funcf1 17923	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
28	27	adantr 485	. . . . . 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 2769	. . . . . . . . . . . 12 ⊢ (Inv‘𝑄) = (Inv‘𝑄)
32	2, 31, 11, 6, 12, 5	isoval 17822	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
33	32	eleq2d 2855	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
34	2, 31, 11, 6, 12	invfun 17821	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35		funfvbrb 7047	. . . . . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴))
39	1, 18, 3, 6, 12, 31, 16	fucinv 18033	. . . . . . . . 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 485	. . . . . . . 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 1160	. . . . . 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 3263	. . . . 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 17823	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
45	44	ralrimiva 3163	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
46	14, 45	jca 520	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))))
47	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝑄 ∈ Cat)
48	6	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐹 ∈ (𝐶 Func 𝐷))
49	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐺 ∈ (𝐶 Func 𝐷))
50		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
51	10	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
52	22	adantr 485	. . . . . 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 485	. . . . . 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 784	. . . . . 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 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
58		fveq2 6882	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
59		fveq2 6882	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
60	58, 59	oveq12d 7429	. . . . . . . 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 2863	. . . . . . 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 3589	. . . . . 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 591	. . . . 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 17848	. . . 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 18034	. . 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 17823	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
67	46, 66	impbida 812	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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 Rel wrel 5667 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 1^st c1st 7984 2^nd c2nd 7985 Basecbs 17269 Catccat 17720 Invcinv 17802 Isociso 17803 Func cfunc 17911 Nat cnat 18001 FuncCat cfuc 18002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-fz 13536 df-struct 17207 df-slot 17242 df-ndx 17254 df-base 17270 df-hom 17334 df-cco 17335 df-cat 17724 df-cid 17725 df-sect 17804 df-inv 17805 df-iso 17806 df-func 17915 df-nat 18003 df-fuc 18004

This theorem is referenced by: yonffthlem 18338

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