fuciso - Metamath Proof Explorer

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Theorem fuciso 17872

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 17857	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
3		fuciso.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4	1, 3	fuchom 17858	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
5		fuciso.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝑄)
6		fuciso.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 17757	. . . . . . . 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 17867	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ Cat)
12		fuciso.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
13	2, 4, 5, 11, 6, 12	isohom 17670	. . . 4 ⊢ (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
14	13	sselda 3931	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15		eqid 2729	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2729	. . . . 5 ⊢ (Inv‘𝐷) = (Inv‘𝐷)
17	10	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ Cat)
18		fuciso.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
19		relfunc 17756	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝐷)
20		1st2ndbr 7968	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
21	19, 6, 20	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
22	18, 15, 21	funcf1 17760	. . . . . . 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 7011	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
25		1st2ndbr 7968	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
26	19, 12, 25	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
27	18, 15, 26	funcf1 17760	. . . . . . 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 7011	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
30		fuciso.j	. . . . 5 ⊢ 𝐽 = (Iso‘𝐷)
31		eqid 2729	. . . . . . . . . . . 12 ⊢ (Inv‘𝑄) = (Inv‘𝑄)
32	2, 31, 11, 6, 12, 5	isoval 17659	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
33	32	eleq2d 2814	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
34	2, 31, 11, 6, 12	invfun 17658	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35		funfvbrb 6978	. . . . . . . . . . 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 17870	. . . . . . . . 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 1144	. . . . . 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 3221	. . . . 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 17660	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥 ∈ 𝐵) → (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))
45	44	ralrimiva 3121	. . 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
51	10	ad2antrr 726	. . . . 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 7011	. . . . 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 7011	. . . . 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 772	. . . . . 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 6816	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
58		fveq2 6816	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐹)‘𝑥) = ((1^st ‘𝐹)‘𝑦))
59		fveq2 6816	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝐺)‘𝑥) = ((1^st ‘𝐺)‘𝑦))
60	58, 59	oveq12d 7358	. . . . . . . 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 2822	. . . . . . 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 3573	. . . . . 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 580	. . . . 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 17685	. . . 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 17871	. . 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 17660	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
67	46, 66	impbida 800	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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 class class class wbr 5088 ↦ cmpt 5169 dom cdm 5613 Rel wrel 5618 Fun wfun 6470 ⟶wf 6472 ‘cfv 6476 (class class class)co 7340 1^st c1st 7913 2^nd c2nd 7914 Basecbs 17107 Catccat 17557 Invcinv 17639 Isociso 17640 Func cfunc 17748 Nat cnat 17838 FuncCat cfuc 17839

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 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074

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 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-om 7791 df-1st 7915 df-2nd 7916 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-er 8616 df-map 8746 df-ixp 8816 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-nn 12117 df-2 12179 df-3 12180 df-4 12181 df-5 12182 df-6 12183 df-7 12184 df-8 12185 df-9 12186 df-n0 12373 df-z 12460 df-dec 12580 df-uz 12724 df-fz 13399 df-struct 17045 df-slot 17080 df-ndx 17092 df-base 17108 df-hom 17172 df-cco 17173 df-cat 17561 df-cid 17562 df-sect 17641 df-inv 17642 df-iso 17643 df-func 17752 df-nat 17840 df-fuc 17841

This theorem is referenced by: yonffthlem 18175

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