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Theorem funcrngcsetc 46286
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 46287, using cofuval2 17773 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 46285, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18037. (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 RngHomo 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 2736 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 funcrngcsetc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
3 eqid 2736 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4 eqid 2736 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
5 funcrngcsetc.u . . . . . 6 (𝜑𝑈 ∈ WUni)
61, 5estrcbas 18012 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
76mpteq1d 5200 . . . . . 6 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8 mpoeq12 7430 . . . . . . 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 18037 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11 df-br 5106 . . . . 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 217 . . . 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 2736 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1513, 14, 5rngcbas 46253 . . . . . 6 (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16 incom 4161 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1715, 16eqtrdi 2792 . . . . 5 (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18 eqid 2736 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
1913, 14, 5, 18rngchomfval 46254 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ ((Base‘𝑅) × (Base‘𝑅))))
201, 5, 17, 19rnghmsubcsetc 46265 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
2112, 20funcres 17782 . . 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 7171 . . . . . 6 (𝑈 ∈ WUni → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
235, 22syl 17 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
24 fvex 6855 . . . . . 6 (Hom ‘𝑅) ∈ V
2524a1i 11 . . . . 5 (𝜑 → (Hom ‘𝑅) ∈ V)
26 mpoexga 8010 . . . . . 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 46251 . . . . 5 (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
2923, 25, 27, 28resfval2 17779 . . . 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 4188 . . . . . . . 8 (𝑈 ∩ Rng) ⊆ 𝑈
3115, 30eqsstrdi 3998 . . . . . . 7 (𝜑 → (Base‘𝑅) ⊆ 𝑈)
3231resmptd 5994 . . . . . 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 5200 . . . . . . 7 (𝜑 → (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
3733, 36eqtr2d 2777 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
3832, 37eqtrd 2776 . . . . 5 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39 funcrngcsetc.g . . . . . 6 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
40 oveq1 7364 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥 RngHomo 𝑦) = (𝑎 RngHomo 𝑦))
4140reseq2d 5937 . . . . . . . 8 (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑎 RngHomo 𝑦)))
42 oveq2 7365 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑎 RngHomo 𝑦) = (𝑎 RngHomo 𝑏))
4342reseq2d 5937 . . . . . . . 8 (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHomo 𝑦)) = ( I ↾ (𝑎 RngHomo 𝑏)))
4441, 43cbvmpov 7452 . . . . . . 7 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHomo 𝑏)))
4544a1i 11 . . . . . 6 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHomo 𝑏))))
4634a1i 11 . . . . . . 7 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝑅))
47 eqidd 2737 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
48 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49 fveq2 6842 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
5048, 49oveqan12rd 7377 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
5150reseq2d 5937 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
5251adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
5334, 31eqsstrid 3992 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
5453sseld 3943 . . . . . . . . . . . . 13 (𝜑 → (𝑎𝐵𝑎𝑈))
5554com12 32 . . . . . . . . . . . 12 (𝑎𝐵 → (𝜑𝑎𝑈))
5655adantr 481 . . . . . . . . . . 11 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
5756impcom 408 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
5853sseld 3943 . . . . . . . . . . . 12 (𝜑 → (𝑏𝐵𝑏𝑈))
5958adantld 491 . . . . . . . . . . 11 (𝜑 → ((𝑎𝐵𝑏𝐵) → 𝑏𝑈))
6059imp 407 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
61 ovexd 7392 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
6261resiexd 7166 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
6347, 52, 57, 60, 62ovmpod 7507 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
6463reseq1d 5936 . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
66 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
67 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
6813, 34, 65, 18, 66, 67rngchom 46255 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHomo 𝑏))
6968reseq2d 5937 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHomo 𝑏)))
70 eqid 2736 . . . . . . . . . . . 12 (Base‘𝑎) = (Base‘𝑎)
71 eqid 2736 . . . . . . . . . . . 12 (Base‘𝑏) = (Base‘𝑏)
7270, 71rnghmf 46187 . . . . . . . . . . 11 (𝑓 ∈ (𝑎 RngHomo 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73 fvex 6855 . . . . . . . . . . . . . 14 (Base‘𝑏) ∈ V
74 fvex 6855 . . . . . . . . . . . . . 14 (Base‘𝑎) ∈ V
7573, 74pm3.2i 471 . . . . . . . . . . . . 13 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
7675a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V))
77 elmapg 8778 . . . . . . . . . . . 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, 78syl5ibr 245 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ (𝑎 RngHomo 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
8079ssrdv 3950 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
8180resabs1d 5968 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHomo 𝑏)) = ( I ↾ (𝑎 RngHomo 𝑏)))
8264, 69, 813eqtrrd 2781 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ (𝑎 RngHomo 𝑏)) = ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
8335, 46, 82mpoeq123dva 7431 . . . . . 6 (𝜑 → (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RngHomo 𝑏))) = (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))))
8439, 45, 833eqtrrd 2781 . . . . 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 4838 . . . 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 2777 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)))
8713, 5, 15, 19rngcval 46250 . . . 4 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
8887oveq1d 7372 . . 3 (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
8921, 86, 883eltr4d 2853 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 5106 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
9189, 90sylibr 233 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cin 3909  cop 4592   class class class wbr 5105  cmpt 5188   I cid 5530  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  WUnicwun 10636  Basecbs 17083  Hom chom 17144  cat cresc 17691   Func cfunc 17740  f cresf 17743  SetCatcsetc 17961  ExtStrCatcestrc 18009  Rngcrng 46162   RngHomo crngh 46173  RngCatcrngc 46245
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wun 10638  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-hom 17157  df-cco 17158  df-0g 17323  df-cat 17548  df-cid 17549  df-homf 17550  df-ssc 17693  df-resc 17694  df-subc 17695  df-func 17744  df-resf 17747  df-setc 17962  df-estrc 18010  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-abl 19565  df-mgp 19897  df-mgmhm 46063  df-rng 46163  df-rnghomo 46175  df-rngc 46247
This theorem is referenced by: (None)
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