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Theorem funcrngcsetc 20640
Description: The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 20641, using cofuval2 17932 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 20639, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18194. (Contributed by AV, 26-Mar-2020.)
Hypotheses
Ref Expression
funcrngcsetc.r 𝑅 = (RngCat‘𝑈)
funcrngcsetc.s 𝑆 = (SetCat‘𝑈)
funcrngcsetc.b 𝐵 = (Base‘𝑅)
funcrngcsetc.u (𝜑𝑈 ∈ WUni)
funcrngcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcrngcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))))
Assertion
Ref Expression
funcrngcsetc (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcrngcsetc
Dummy variables 𝑎 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 funcrngcsetc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
3 eqid 2737 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4 eqid 2737 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
5 funcrngcsetc.u . . . . . 6 (𝜑𝑈 ∈ WUni)
61, 5estrcbas 18169 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
76mpteq1d 5237 . . . . . 6 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8 mpoeq12 7506 . . . . . . 7 ((𝑈 = (Base‘(ExtStrCat‘𝑈)) ∧ 𝑈 = (Base‘(ExtStrCat‘𝑈))) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑦 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
96, 6, 8syl2anc 584 . . . . . 6 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑦 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
101, 2, 3, 4, 5, 7, 9funcestrcsetc 18194 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11 df-br 5144 . . . . 5 ((𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ↔ ⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
1210, 11sylib 218 . . . 4 (𝜑 → ⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
13 funcrngcsetc.r . . . . . . 7 𝑅 = (RngCat‘𝑈)
14 eqid 2737 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1513, 14, 5rngcbas 20621 . . . . . 6 (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16 incom 4209 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1715, 16eqtrdi 2793 . . . . 5 (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18 eqid 2737 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
1913, 14, 5, 18rngchomfval 20622 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
201, 5, 17, 19rnghmsubcsetc 20633 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
2112, 20funcres 17941 . . 3 (𝜑 → (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)) ∈ (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
22 mptexg 7241 . . . . . 6 (𝑈 ∈ WUni → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
235, 22syl 17 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
24 fvex 6919 . . . . . 6 (Hom ‘𝑅) ∈ V
2524a1i 11 . . . . 5 (𝜑 → (Hom ‘𝑅) ∈ V)
26 mpoexga 8102 . . . . . 6 ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
275, 5, 26syl2anc 584 . . . . 5 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
2815, 19rnghmresfn 20619 . . . . 5 (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
2923, 25, 27, 28resfval2 17938 . . . 4 (𝜑 → (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)) = ⟨((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)), (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))⟩)
30 inss1 4237 . . . . . . . 8 (𝑈 ∩ Rng) ⊆ 𝑈
3115, 30eqsstrdi 4028 . . . . . . 7 (𝜑 → (Base‘𝑅) ⊆ 𝑈)
3231resmptd 6058 . . . . . 6 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
33 funcrngcsetc.f . . . . . . 7 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
34 funcrngcsetc.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3534a1i 11 . . . . . . . 8 (𝜑𝐵 = (Base‘𝑅))
3635mpteq1d 5237 . . . . . . 7 (𝜑 → (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
3733, 36eqtr2d 2778 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
3832, 37eqtrd 2777 . . . . 5 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39 funcrngcsetc.g . . . . . 6 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))))
40 oveq1 7438 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥 RngHom 𝑦) = (𝑎 RngHom 𝑦))
4140reseq2d 5997 . . . . . . . 8 (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑦)))
42 oveq2 7439 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑎 RngHom 𝑦) = (𝑎 RngHom 𝑏))
4342reseq2d 5997 . . . . . . . 8 (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑏)))
4441, 43cbvmpov 7528 . . . . . . 7 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHom 𝑏)))
4544a1i 11 . . . . . 6 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHom 𝑏))))
4634a1i 11 . . . . . . 7 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝑅))
47 eqidd 2738 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
48 fveq2 6906 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
5048, 49oveqan12rd 7451 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
5150reseq2d 5997 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
5251adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
5334, 31eqsstrid 4022 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
5453sseld 3982 . . . . . . . . . . . . 13 (𝜑 → (𝑎𝐵𝑎𝑈))
5554com12 32 . . . . . . . . . . . 12 (𝑎𝐵 → (𝜑𝑎𝑈))
5655adantr 480 . . . . . . . . . . 11 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
5756impcom 407 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
5853sseld 3982 . . . . . . . . . . . 12 (𝜑 → (𝑏𝐵𝑏𝑈))
5958adantld 490 . . . . . . . . . . 11 (𝜑 → ((𝑎𝐵𝑏𝐵) → 𝑏𝑈))
6059imp 406 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
61 ovexd 7466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
6261resiexd 7236 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
6347, 52, 57, 60, 62ovmpod 7585 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
6463reseq1d 5996 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)))
655adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
66 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
67 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
6813, 34, 65, 18, 66, 67rngchom 20623 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHom 𝑏))
6968reseq2d 5997 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHom 𝑏)))
70 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑎) = (Base‘𝑎)
71 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑏) = (Base‘𝑏)
7270, 71rnghmf 20448 . . . . . . . . . . 11 (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73 fvex 6919 . . . . . . . . . . . . . 14 (Base‘𝑏) ∈ V
74 fvex 6919 . . . . . . . . . . . . . 14 (Base‘𝑎) ∈ V
7573, 74pm3.2i 470 . . . . . . . . . . . . 13 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
7675a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V))
77 elmapg 8879 . . . . . . . . . . . 12 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
7876, 77syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
7972, 78imbitrrid 246 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
8079ssrdv 3989 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
8180resabs1d 6026 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHom 𝑏)) = ( I ↾ (𝑎 RngHom 𝑏)))
8264, 69, 813eqtrrd 2782 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ (𝑎 RngHom 𝑏)) = ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
8335, 46, 82mpoeq123dva 7507 . . . . . 6 (𝜑 → (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHom 𝑏))) = (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))))
8439, 45, 833eqtrrd 2782 . . . . 5 (𝜑 → (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))) = 𝐺)
8538, 84opeq12d 4881 . . . 4 (𝜑 → ⟨((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)), (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))⟩ = ⟨𝐹, 𝐺⟩)
8629, 85eqtr2d 2778 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)))
8713, 5, 15, 19rngcval 20618 . . . 4 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
8887oveq1d 7446 . . 3 (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
8921, 86, 883eltr4d 2856 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 5144 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
9189, 90sylibr 234 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866  WUnicwun 10740  Basecbs 17247  Hom chom 17308  cat cresc 17852   Func cfunc 17899  f cresf 17902  SetCatcsetc 18120  ExtStrCatcestrc 18166  Rngcrng 20149   RngHom crnghm 20434  RngCatcrngc 20616
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-wun 10742  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-hom 17321  df-cco 17322  df-0g 17486  df-cat 17711  df-cid 17712  df-homf 17713  df-ssc 17854  df-resc 17855  df-subc 17856  df-func 17903  df-resf 17906  df-setc 18121  df-estrc 18167  df-mgm 18653  df-mgmhm 18705  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-ghm 19231  df-abl 19801  df-mgp 20138  df-rng 20150  df-rnghm 20436  df-rngc 20617
This theorem is referenced by: (None)
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