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Theorem fuciso 16575
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 16560 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
3 fuciso.n . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
41, 3fuchom 16561 . . . . 5 𝑁 = (Hom ‘𝑄)
5 fuciso.i . . . . 5 𝐼 = (Iso‘𝑄)
6 fuciso.f . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 16463 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simpld 475 . . . . . 6 (𝜑𝐶 ∈ Cat)
108simprd 479 . . . . . 6 (𝜑𝐷 ∈ Cat)
111, 9, 10fuccat 16570 . . . . 5 (𝜑𝑄 ∈ Cat)
12 fuciso.g . . . . 5 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
132, 4, 5, 11, 6, 12isohom 16376 . . . 4 (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
1413sselda 3588 . . 3 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15 eqid 2621 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2621 . . . . 5 (Inv‘𝐷) = (Inv‘𝐷)
1710ad2antrr 761 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → 𝐷 ∈ Cat)
18 fuciso.b . . . . . . . 8 𝐵 = (Base‘𝐶)
19 relfunc 16462 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
20 1st2ndbr 7177 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2119, 6, 20sylancr 694 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2218, 15, 21funcf1 16466 . . . . . . 7 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2322adantr 481 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (1st𝐹):𝐵⟶(Base‘𝐷))
2423ffvelrnda 6325 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
25 1st2ndbr 7177 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2619, 12, 25sylancr 694 . . . . . . . 8 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2718, 15, 26funcf1 16466 . . . . . . 7 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2827adantr 481 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (1st𝐺):𝐵⟶(Base‘𝐷))
2928ffvelrnda 6325 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
30 fuciso.j . . . . 5 𝐽 = (Iso‘𝐷)
31 eqid 2621 . . . . . . . . . . . 12 (Inv‘𝑄) = (Inv‘𝑄)
322, 31, 11, 6, 12, 5isoval 16365 . . . . . . . . . . 11 (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
3332eleq2d 2684 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
342, 31, 11, 6, 12invfun 16364 . . . . . . . . . . 11 (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35 funfvbrb 6296 . . . . . . . . . . 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 501 . . . . . . . 8 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴))
391, 18, 3, 6, 12, 31, 16fucinv 16573 . . . . . . . . 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 481 . . . . . . . 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 1073 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
4342r19.21bi 2928 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
4415, 16, 17, 24, 29, 30, 43inviso1 16366 . . . 4 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
4544ralrimiva 2962 . . 3 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
4614, 45jca 554 . 2 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))))
4711adantr 481 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝑄 ∈ Cat)
486adantr 481 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐹 ∈ (𝐶 Func 𝐷))
4912adantr 481 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐺 ∈ (𝐶 Func 𝐷))
50 simprl 793 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
51 simprr 795 . . . . . . 7 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
52 fveq2 6158 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
53 fveq2 6158 . . . . . . . . . 10 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
54 fveq2 6158 . . . . . . . . . 10 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
5553, 54oveq12d 6633 . . . . . . . . 9 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
5652, 55eleq12d 2692 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ↔ (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))))
5756rspccva 3298 . . . . . . 7 ((∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
5851, 57sylan 488 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
5910ad2antrr 761 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → 𝐷 ∈ Cat)
6022adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → (1st𝐹):𝐵⟶(Base‘𝐷))
6160ffvelrnda 6325 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
6227adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → (1st𝐺):𝐵⟶(Base‘𝐷))
6362ffvelrnda 6325 . . . . . . 7 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
6415, 16, 59, 61, 63, 30isoval 16365 . . . . . 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 2700 . . . . 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 16364 . . . . . 6 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → Fun (((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦)))
67 funfvbrb 6296 . . . . . 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 16574 . . 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 16366 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
7246, 71impbida 876 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908   class class class wbr 4623  cmpt 4683  dom cdm 5084  Rel wrel 5089  Fun wfun 5851  wf 5853  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Basecbs 15800  Catccat 16265  Invcinv 16345  Isociso 16346   Func cfunc 16454   Nat cnat 16541   FuncCat cfuc 16542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-hom 15906  df-cco 15907  df-cat 16269  df-cid 16270  df-sect 16347  df-inv 16348  df-iso 16349  df-func 16458  df-nat 16543  df-fuc 16544
This theorem is referenced by:  yonffthlem  16862
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