fuciso - Metamath Proof Explorer

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Theorem fuciso 17916

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 17901	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 17902	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
5		fuciso.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝑄)
6		fuciso.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 17801	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9	8	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
10	8	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
11	1, 9, 10	fuccat 17911	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
12		fuciso.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
13	2, 4, 5, 11, 6, 12	isohom 17714	. . . 4 ⊢ (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
14	13	sselda 3935	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15		eqid 2737	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2737	. . . . 5 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17	10	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
18		fuciso.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
19		relfunc 17800	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
20		1st2ndbr 7998	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
21	19, 6, 20	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
22	18, 15, 21	funcf1 17804	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
23	22	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (1^st ‘𝐹):𝐵⟶(Base‘𝐷))
24	23	ffvelcdmda 7040	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
25		1st2ndbr 7998	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
26	19, 12, 25	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
27	18, 15, 26	funcf1 17804	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
28	27	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (1^st ‘𝐺):𝐵⟶(Base‘𝐷))
29	28	ffvelcdmda 7040	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30		fuciso.j	. . . . 5 ⊢ 𝐽 = (Iso‘𝐷)
31		eqid 2737	. . . . . . . . . . . 12 ⊢ (Inv‘𝑄) = (Inv‘𝑄)
32	2, 31, 11, 6, 12, 5	isoval 17703	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
33	32	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
34	2, 31, 11, 6, 12	invfun 17702	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35		funfvbrb 7007	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
37	33, 36	bitrd 279	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
38	37	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴))
39	1, 18, 3, 6, 12, 31, 16	fucinv 17914	. . . . . . . . 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 480	. . . . . . . 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 232	. . . . . . 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 1145	. . . . . 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 3230	. . . . 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 17704	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
45	44	ralrimiva 3130	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
46	14, 45	jca 511	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥))))
47	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝑄 ∈ Cat)
48	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐹 ∈ (𝐶 Func 𝐷))
49	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐺 ∈ (𝐶 Func 𝐷))
50		simprl 771	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
51	10	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
52	22	adantr 480	. . . . . 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 7040	. . . . 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 480	. . . . . 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 7040	. . . . 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 773	. . . . . 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 6844	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
58		fveq2 6844	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
59		fveq2 6844	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
60	58, 59	oveq12d 7388	. . . . . . . 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 2831	. . . . . . 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 3577	. . . . . 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 581	. . . . 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 17729	. . . 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 17915	. . 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 17704	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
67	46, 66	impbida 801	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5634 Rel wrel 5639 Fun wfun 6496 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 1^st c1st 7943 2^nd c2nd 7944 Basecbs 17150 Catccat 17601 Invcinv 17683 Isociso 17684 Func cfunc 17792 Nat cnat 17882 FuncCat cfuc 17883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-map 8779 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-fz 13438 df-struct 17088 df-slot 17123 df-ndx 17135 df-base 17151 df-hom 17215 df-cco 17216 df-cat 17605 df-cid 17606 df-sect 17685 df-inv 17686 df-iso 17687 df-func 17796 df-nat 17884 df-fuc 17885

This theorem is referenced by: yonffthlem 18219

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