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Theorem fuciso 16841
 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 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
fuciso.i 𝐼 = (Iso‘𝑄)
fuciso.j 𝐽 = (Iso‘𝐷)
Assertion
Ref Expression
fuciso (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘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.
StepHypRef Expression
1 fuciso.q . . . . . 6 𝑄 = (𝐶 FuncCat 𝐷)
21fucbas 16826 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
3 fuciso.n . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
41, 3fuchom 16827 . . . . 5 𝑁 = (Hom ‘𝑄)
5 fuciso.i . . . . 5 𝐼 = (Iso‘𝑄)
6 fuciso.f . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 16729 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simpld 476 . . . . . 6 (𝜑𝐶 ∈ Cat)
108simprd 477 . . . . . 6 (𝜑𝐷 ∈ Cat)
111, 9, 10fuccat 16836 . . . . 5 (𝜑𝑄 ∈ Cat)
12 fuciso.g . . . . 5 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
132, 4, 5, 11, 6, 12isohom 16642 . . . 4 (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
1413sselda 3750 . . 3 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15 eqid 2770 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2770 . . . . 5 (Inv‘𝐷) = (Inv‘𝐷)
1710ad2antrr 697 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → 𝐷 ∈ Cat)
18 fuciso.b . . . . . . . 8 𝐵 = (Base‘𝐶)
19 relfunc 16728 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
20 1st2ndbr 7365 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2119, 6, 20sylancr 567 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2218, 15, 21funcf1 16732 . . . . . . 7 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2322adantr 466 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (1st𝐹):𝐵⟶(Base‘𝐷))
2423ffvelrnda 6502 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
25 1st2ndbr 7365 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2619, 12, 25sylancr 567 . . . . . . . 8 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2718, 15, 26funcf1 16732 . . . . . . 7 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2827adantr 466 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (1st𝐺):𝐵⟶(Base‘𝐷))
2928ffvelrnda 6502 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
30 fuciso.j . . . . 5 𝐽 = (Iso‘𝐷)
31 eqid 2770 . . . . . . . . . . . 12 (Inv‘𝑄) = (Inv‘𝑄)
322, 31, 11, 6, 12, 5isoval 16631 . . . . . . . . . . 11 (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
3332eleq2d 2835 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
342, 31, 11, 6, 12invfun 16630 . . . . . . . . . . 11 (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35 funfvbrb 6473 . . . . . . . . . . 11 (Fun (𝐹(Inv‘𝑄)𝐺) → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
3634, 35syl 17 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
3733, 36bitrd 268 . . . . . . . . 9 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
3837biimpa 462 . . . . . . . 8 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴))
391, 18, 3, 6, 12, 31, 16fucinv 16839 . . . . . . . . 9 (𝜑 → (𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))))
4039adantr 466 . . . . . . . 8 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))))
4138, 40mpbid 222 . . . . . . 7 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥)))
4241simp3d 1137 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
4342r19.21bi 3080 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
4415, 16, 17, 24, 29, 30, 43inviso1 16632 . . . 4 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
4544ralrimiva 3114 . . 3 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
4614, 45jca 495 . 2 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))))
4711adantr 466 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝑄 ∈ Cat)
486adantr 466 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐹 ∈ (𝐶 Func 𝐷))
4912adantr 466 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐺 ∈ (𝐶 Func 𝐷))
50 simprl 746 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
51 simprr 748 . . . . . . 7 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
52 fveq2 6332 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
53 fveq2 6332 . . . . . . . . . 10 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
54 fveq2 6332 . . . . . . . . . 10 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
5553, 54oveq12d 6810 . . . . . . . . 9 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
5652, 55eleq12d 2843 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ↔ (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))))
5756rspccva 3457 . . . . . . 7 ((∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
5851, 57sylan 561 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
5910ad2antrr 697 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → 𝐷 ∈ Cat)
6022adantr 466 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → (1st𝐹):𝐵⟶(Base‘𝐷))
6160ffvelrnda 6502 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
6227adantr 466 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → (1st𝐺):𝐵⟶(Base‘𝐷))
6362ffvelrnda 6502 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
6415, 16, 59, 61, 63, 30isoval 16631 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)) = dom (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)))
6558, 64eleqtrd 2851 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ dom (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)))
6615, 16, 59, 61, 63invfun 16630 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → Fun (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)))
67 funfvbrb 6473 . . . . . 6 (Fun (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)) → ((𝐴𝑦) ∈ dom (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)) ↔ (𝐴𝑦)(((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))((((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))‘(𝐴𝑦))))
6866, 67syl 17 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((𝐴𝑦) ∈ dom (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)) ↔ (𝐴𝑦)(((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))((((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))‘(𝐴𝑦))))
6965, 68mpbid 222 . . . 4 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (𝐴𝑦)(((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))((((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))‘(𝐴𝑦)))
701, 18, 3, 48, 49, 31, 16, 50, 69invfuc 16840 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴(𝐹(Inv‘𝑄)𝐺)(𝑦𝐵 ↦ ((((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))‘(𝐴𝑦))))
712, 31, 47, 48, 49, 5, 70inviso1 16632 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
7246, 71impbida 794 1 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∀wral 3060   class class class wbr 4784   ↦ cmpt 4861  dom cdm 5249  Rel wrel 5254  Fun wfun 6025  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  Basecbs 16063  Catccat 16531  Invcinv 16611  Isociso 16612   Func cfunc 16720   Nat cnat 16807   FuncCat cfuc 16808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-hom 16173  df-cco 16174  df-cat 16535  df-cid 16536  df-sect 16613  df-inv 16614  df-iso 16615  df-func 16724  df-nat 16809  df-fuc 16810 This theorem is referenced by:  yonffthlem  17129
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