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Theorem funcrngcsetc 42508
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 42509, using cofuval2 16748 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 42507, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16990. (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 2760 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 funcrngcsetc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
3 eqid 2760 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4 eqid 2760 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
5 funcrngcsetc.u . . . . . 6 (𝜑𝑈 ∈ WUni)
61, 5estrcbas 16966 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
76mpteq1d 4890 . . . . . 6 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8 mpt2eq12 6880 . . . . . . 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 696 . . . . . 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 16990 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
11 df-br 4805 . . . . 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 208 . . . 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 2760 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1513, 14, 5rngcbas 42475 . . . . . 6 (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16 incom 3948 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
1715, 16syl6eq 2810 . . . . 5 (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18 eqid 2760 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
1913, 14, 5, 18rngchomfval 42476 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ ((Base‘𝑅) × (Base‘𝑅))))
201, 5, 17, 19rnghmsubcsetc 42487 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
2112, 20funcres 16757 . . 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 6648 . . . . . 6 (𝑈 ∈ WUni → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
235, 22syl 17 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
24 fvex 6362 . . . . . 6 (Hom ‘𝑅) ∈ V
2524a1i 11 . . . . 5 (𝜑 → (Hom ‘𝑅) ∈ V)
26 mpt2exga 7414 . . . . . 6 ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ∈ V)
275, 5, 26syl2anc 696 . . . . 5 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ∈ V)
2815, 19rnghmresfn 42473 . . . . 5 (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
2923, 25, 27, 28resfval2 16754 . . . 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 3976 . . . . . . . 8 (𝑈 ∩ Rng) ⊆ 𝑈
3115, 30syl6eqss 3796 . . . . . . 7 (𝜑 → (Base‘𝑅) ⊆ 𝑈)
3231resmptd 5610 . . . . . 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 4890 . . . . . . 7 (𝜑 → (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
3733, 36eqtr2d 2795 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
3832, 37eqtrd 2794 . . . . 5 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39 funcrngcsetc.g . . . . . 6 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
40 oveq1 6820 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥 RngHomo 𝑦) = (𝑎 RngHomo 𝑦))
4140reseq2d 5551 . . . . . . . 8 (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑎 RngHomo 𝑦)))
42 oveq2 6821 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑎 RngHomo 𝑦) = (𝑎 RngHomo 𝑏))
4342reseq2d 5551 . . . . . . . 8 (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHomo 𝑦)) = ( I ↾ (𝑎 RngHomo 𝑏)))
4441, 43cbvmpt2v 6900 . . . . . . 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 2761 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) = (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
48 fveq2 6352 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49 fveq2 6352 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
5048, 49oveqan12rd 6833 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
5150reseq2d 5551 . . . . . . . . . . 11 ((𝑥 = 𝑎𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
5251adantl 473 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐵𝑏𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
5334, 31syl5eqss 3790 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
5453sseld 3743 . . . . . . . . . . . . 13 (𝜑 → (𝑎𝐵𝑎𝑈))
5554com12 32 . . . . . . . . . . . 12 (𝑎𝐵 → (𝜑𝑎𝑈))
5655adantr 472 . . . . . . . . . . 11 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
5756impcom 445 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
5853sseld 3743 . . . . . . . . . . . 12 (𝜑 → (𝑏𝐵𝑏𝑈))
5958adantld 484 . . . . . . . . . . 11 (𝜑 → ((𝑎𝐵𝑏𝐵) → 𝑏𝑈))
6059imp 444 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
61 ovexd 6843 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
6261resiexd 6644 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ∈ V)
6347, 52, 57, 60, 62ovmpt2d 6953 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
6463reseq1d 5550 . . . . . . . 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 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
66 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
67 simprr 813 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
6813, 34, 65, 18, 66, 67rngchom 42477 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHomo 𝑏))
6968reseq2d 5551 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎 RngHomo 𝑏)))
70 eqid 2760 . . . . . . . . . . . 12 (Base‘𝑎) = (Base‘𝑎)
71 eqid 2760 . . . . . . . . . . . 12 (Base‘𝑏) = (Base‘𝑏)
7270, 71rnghmf 42409 . . . . . . . . . . 11 (𝑓 ∈ (𝑎 RngHomo 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73 fvex 6362 . . . . . . . . . . . . . 14 (Base‘𝑏) ∈ V
74 fvex 6362 . . . . . . . . . . . . . 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 8036 . . . . . . . . . . . 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 236 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ (𝑎 RngHomo 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
8079ssrdv 3750 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
8180resabs1d 5586 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎 RngHomo 𝑏)) = ( I ↾ (𝑎 RngHomo 𝑏)))
8264, 69, 813eqtrrd 2799 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ (𝑎 RngHomo 𝑏)) = ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
8335, 46, 82mpt2eq123dva 6881 . . . . . 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 2799 . . . . 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 4561 . . . 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 2795 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)))
8713, 5, 15, 19rngcval 42472 . . . 4 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
8887oveq1d 6828 . . 3 (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
8921, 86, 883eltr4d 2854 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 4805 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
9189, 90sylibr 224 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  cop 4327   class class class wbr 4804  cmpt 4881   I cid 5173  cres 5268  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  WUnicwun 9714  Basecbs 16059  Hom chom 16154  cat cresc 16669   Func cfunc 16715  f cresf 16718  SetCatcsetc 16926  ExtStrCatcestrc 16963  Rngcrng 42384   RngHomo crngh 42395  RngCatcrngc 42467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-wun 9716  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-hom 16168  df-cco 16169  df-0g 16304  df-cat 16530  df-cid 16531  df-homf 16532  df-ssc 16671  df-resc 16672  df-subc 16673  df-func 16719  df-resf 16722  df-setc 16927  df-estrc 16964  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-grp 17626  df-ghm 17859  df-abl 18396  df-mgp 18690  df-mgmhm 42289  df-rng0 42385  df-rnghomo 42397  df-rngc 42469
This theorem is referenced by: (None)
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