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Theorem funcringcsetc 41339
Description: The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
Hypotheses
Ref Expression
funcringcsetc.r 𝑅 = (RingCat‘𝑈)
funcringcsetc.s 𝑆 = (SetCat‘𝑈)
funcringcsetc.b 𝐵 = (Base‘𝑅)
funcringcsetc.u (𝜑𝑈 ∈ WUni)
funcringcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetc (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅   𝑥,𝑆   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑅(𝑦)   𝑆(𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcringcsetc
Dummy variables 𝑎 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . 6 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 funcringcsetc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
3 eqid 2621 . . . . . 6 (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4 eqid 2621 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
5 funcringcsetc.u . . . . . 6 (𝜑𝑈 ∈ WUni)
61, 5estrcbas 16689 . . . . . . 7 (𝜑𝑈 = (Base‘(ExtStrCat‘𝑈)))
76mpteq1d 4700 . . . . . 6 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8 mpt2eq12 6671 . . . . . . 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 692 . . . . . 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 16713 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
11 df-br 4616 . . . . 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 funcringcsetc.r . . . . . . 7 𝑅 = (RingCat‘𝑈)
14 eqid 2621 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
1513, 14, 5ringcbas 41315 . . . . . 6 (𝜑 → (Base‘𝑅) = (𝑈 ∩ Ring))
16 incom 3785 . . . . . 6 (𝑈 ∩ Ring) = (Ring ∩ 𝑈)
1715, 16syl6eq 2671 . . . . 5 (𝜑 → (Base‘𝑅) = (Ring ∩ 𝑈))
18 eqid 2621 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
1913, 14, 5, 18ringchomfval 41316 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RingHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
201, 5, 17, 19rhmsubcsetc 41327 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
2112, 20funcres 16480 . . 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 6441 . . . . . 6 (𝑈 ∈ WUni → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
235, 22syl 17 . . . . 5 (𝜑 → (𝑥𝑈 ↦ (Base‘𝑥)) ∈ V)
24 fvex 6160 . . . . . 6 (Hom ‘𝑅) ∈ V
2524a1i 11 . . . . 5 (𝜑 → (Hom ‘𝑅) ∈ V)
26 mpt2exga 7194 . . . . . 6 ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ∈ V)
275, 5, 26syl2anc 692 . . . . 5 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) ∈ V)
2815, 19rhmresfn 41313 . . . . 5 (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
2923, 25, 27, 28resfval2 16477 . . . 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 3813 . . . . . . . 8 (𝑈 ∩ Ring) ⊆ 𝑈
3115, 30syl6eqss 3636 . . . . . . 7 (𝜑 → (Base‘𝑅) ⊆ 𝑈)
3231resmptd 5413 . . . . . 6 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
33 funcringcsetc.f . . . . . . 7 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
34 funcringcsetc.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
3534a1i 11 . . . . . . . 8 (𝜑𝐵 = (Base‘𝑅))
3635mpteq1d 4700 . . . . . . 7 (𝜑 → (𝑥𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
3733, 36eqtr2d 2656 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
3832, 37eqtrd 2655 . . . . 5 (𝜑 → ((𝑥𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39 funcringcsetc.g . . . . . 6 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
40 oveq1 6614 . . . . . . . . 9 (𝑥 = 𝑎 → (𝑥 RingHom 𝑦) = (𝑎 RingHom 𝑦))
4140reseq2d 5358 . . . . . . . 8 (𝑥 = 𝑎 → ( I ↾ (𝑥 RingHom 𝑦)) = ( I ↾ (𝑎 RingHom 𝑦)))
42 oveq2 6615 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑎 RingHom 𝑦) = (𝑎 RingHom 𝑏))
4342reseq2d 5358 . . . . . . . 8 (𝑦 = 𝑏 → ( I ↾ (𝑎 RingHom 𝑦)) = ( I ↾ (𝑎 RingHom 𝑏)))
4441, 43cbvmpt2v 6691 . . . . . . 7 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RingHom 𝑏)))
4544a1i 11 . . . . . 6 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RingHom 𝑏))))
4634a1i 11 . . . . . . 7 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝑅))
47 eqidd 2622 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) = (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))))
48 fveq2 6150 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49 fveq2 6150 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
5048, 49oveqan12rd 6627 . . . . . . . . . . . 12 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
5150reseq2d 5358 . . . . . . . . . . 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, 31syl5eqss 3630 . . . . . . . . . . . . . 14 (𝜑𝐵𝑈)
5453sseld 3583 . . . . . . . . . . . . 13 (𝜑 → (𝑎𝐵𝑎𝑈))
5554com12 32 . . . . . . . . . . . 12 (𝑎𝐵 → (𝜑𝑎𝑈))
5655adantr 481 . . . . . . . . . . 11 ((𝑎𝐵𝑏𝐵) → (𝜑𝑎𝑈))
5756impcom 446 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝑈)
5853sseld 3583 . . . . . . . . . . . 12 (𝜑 → (𝑏𝐵𝑏𝑈))
5958adantld 483 . . . . . . . . . . 11 (𝜑 → ((𝑎𝐵𝑏𝐵) → 𝑏𝑈))
6059imp 445 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝑈)
61 ovex 6635 . . . . . . . . . . . 12 ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V
6261a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
6362resiexd 6437 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ∈ V)
6447, 52, 57, 60, 63ovmpt2d 6744 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) = ( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
6564reseq1d 5357 . . . . . . . 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)))
665adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑈 ∈ WUni)
67 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
68 simprr 795 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
6913, 34, 66, 18, 67, 68ringchom 41317 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RingHom 𝑏))
7069reseq2d 5358 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎 RingHom 𝑏)))
71 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑎) = (Base‘𝑎)
72 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑏) = (Base‘𝑏)
7371, 72rhmf 18650 . . . . . . . . . . 11 (𝑓 ∈ (𝑎 RingHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
74 fvex 6160 . . . . . . . . . . . . . 14 (Base‘𝑏) ∈ V
75 fvex 6160 . . . . . . . . . . . . . 14 (Base‘𝑎) ∈ V
7674, 75pm3.2i 471 . . . . . . . . . . . . 13 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
7776a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V))
78 elmapg 7818 . . . . . . . . . . . 12 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
7977, 78syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
8073, 79syl5ibr 236 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑓 ∈ (𝑎 RingHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
8180ssrdv 3590 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
8281resabs1d 5389 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (( I ↾ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))) ↾ (𝑎 RingHom 𝑏)) = ( I ↾ (𝑎 RingHom 𝑏)))
8365, 70, 823eqtrrd 2660 . . . . . . 7 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ( I ↾ (𝑎 RingHom 𝑏)) = ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
8435, 46, 83mpt2eq123dva 6672 . . . . . 6 (𝜑 → (𝑎𝐵, 𝑏𝐵 ↦ ( I ↾ (𝑎 RingHom 𝑏))) = (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))))
8539, 45, 843eqtrrd 2660 . . . . 5 (𝜑 → (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))) = 𝐺)
8638, 85opeq12d 4380 . . . 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⟩)
8729, 86eqtr2d 2656 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑥𝑈 ↦ (Base‘𝑥)), (𝑥𝑈, 𝑦𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)))
8813, 5, 15, 19ringcval 41312 . . . 4 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
8988oveq1d 6622 . . 3 (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
9021, 87, 893eltr4d 2713 . 2 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
91 df-br 4616 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
9290, 91sylibr 224 1 (𝜑𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3555  cop 4156   class class class wbr 4615  cmpt 4675   I cid 4986  cres 5078  wf 5845  cfv 5849  (class class class)co 6607  cmpt2 6609  𝑚 cmap 7805  WUnicwun 9469  Basecbs 15784  Hom chom 15876  cat cresc 16392   Func cfunc 16438  f cresf 16441  SetCatcsetc 16649  ExtStrCatcestrc 16686  Ringcrg 18471   RingHom crh 18636  RingCatcringc 41307
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-wun 9471  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-dec 11441  df-uz 11635  df-fz 12272  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-hom 15890  df-cco 15891  df-0g 16026  df-cat 16253  df-cid 16254  df-homf 16255  df-ssc 16394  df-resc 16395  df-subc 16396  df-func 16442  df-resf 16445  df-setc 16650  df-estrc 16687  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-mhm 17259  df-grp 17349  df-ghm 17582  df-mgp 18414  df-ur 18426  df-ring 18473  df-rnghom 18639  df-ringc 41309
This theorem is referenced by: (None)
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