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Theorem funcringcsetc 47018
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 = (RingCatβ€˜π‘ˆ)
funcringcsetc.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetc.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetc.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetc.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcringcsetc.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetc (πœ‘ β†’ 𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   π‘₯,𝐡,𝑦   π‘₯,𝑅   π‘₯,𝑆   π‘₯,π‘ˆ,𝑦   πœ‘,π‘₯,𝑦
Allowed substitution hints:   𝑅(𝑦)   𝑆(𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem funcringcsetc
Dummy variables π‘Ž 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . 6 (ExtStrCatβ€˜π‘ˆ) = (ExtStrCatβ€˜π‘ˆ)
2 funcringcsetc.s . . . . . 6 𝑆 = (SetCatβ€˜π‘ˆ)
3 eqid 2732 . . . . . 6 (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))
4 eqid 2732 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
5 funcringcsetc.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ WUni)
61, 5estrcbas 18078 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)))
76mpteq1d 5243 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ (Baseβ€˜π‘₯)))
8 mpoeq12 7484 . . . . . . 7 ((π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ∧ π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))) β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) = (π‘₯ ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)), 𝑦 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
96, 6, 8syl2anc 584 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) = (π‘₯ ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)), 𝑦 ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
101, 2, 3, 4, 5, 7, 9funcestrcsetc 18103 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯))((ExtStrCatβ€˜π‘ˆ) Func 𝑆)(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
11 df-br 5149 . . . . 5 ((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯))((ExtStrCatβ€˜π‘ˆ) Func 𝑆)(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) ↔ ⟨(π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)), (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))⟩ ∈ ((ExtStrCatβ€˜π‘ˆ) Func 𝑆))
1210, 11sylib 217 . . . 4 (πœ‘ β†’ ⟨(π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)), (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))⟩ ∈ ((ExtStrCatβ€˜π‘ˆ) Func 𝑆))
13 funcringcsetc.r . . . . . . 7 𝑅 = (RingCatβ€˜π‘ˆ)
14 eqid 2732 . . . . . . 7 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1513, 14, 5ringcbas 46994 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘…) = (π‘ˆ ∩ Ring))
16 incom 4201 . . . . . 6 (π‘ˆ ∩ Ring) = (Ring ∩ π‘ˆ)
1715, 16eqtrdi 2788 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘…) = (Ring ∩ π‘ˆ))
18 eqid 2732 . . . . . 6 (Hom β€˜π‘…) = (Hom β€˜π‘…)
1913, 14, 5, 18ringchomfval 46995 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) = ( RingHom β†Ύ ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…))))
201, 5, 17, 19rhmsubcsetc 47006 . . . 4 (πœ‘ β†’ (Hom β€˜π‘…) ∈ (Subcatβ€˜(ExtStrCatβ€˜π‘ˆ)))
2112, 20funcres 17848 . . 3 (πœ‘ β†’ (⟨(π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)), (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))⟩ β†Ύf (Hom β€˜π‘…)) ∈ (((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)) Func 𝑆))
22 mptexg 7225 . . . . . 6 (π‘ˆ ∈ WUni β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) ∈ V)
235, 22syl 17 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) ∈ V)
24 fvex 6904 . . . . . 6 (Hom β€˜π‘…) ∈ V
2524a1i 11 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) ∈ V)
26 mpoexga 8066 . . . . . 6 ((π‘ˆ ∈ WUni ∧ π‘ˆ ∈ WUni) β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) ∈ V)
275, 5, 26syl2anc 584 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) ∈ V)
2815, 19rhmresfn 46992 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) Fn ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…)))
2923, 25, 27, 28resfval2 17845 . . . 4 (πœ‘ β†’ (⟨(π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)), (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))⟩ β†Ύf (Hom β€˜π‘…)) = ⟨((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) β†Ύ (Baseβ€˜π‘…)), (π‘Ž ∈ (Baseβ€˜π‘…), 𝑏 ∈ (Baseβ€˜π‘…) ↦ ((π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏)))⟩)
30 inss1 4228 . . . . . . . 8 (π‘ˆ ∩ Ring) βŠ† π‘ˆ
3115, 30eqsstrdi 4036 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) βŠ† π‘ˆ)
3231resmptd 6040 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) β†Ύ (Baseβ€˜π‘…)) = (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)))
33 funcringcsetc.f . . . . . . 7 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
34 funcringcsetc.b . . . . . . . . 9 𝐡 = (Baseβ€˜π‘…)
3534a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))
3635mpteq1d 5243 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)))
3733, 36eqtr2d 2773 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)) = 𝐹)
3832, 37eqtrd 2772 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) β†Ύ (Baseβ€˜π‘…)) = 𝐹)
39 funcringcsetc.g . . . . . 6 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
40 oveq1 7418 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ (π‘₯ RingHom 𝑦) = (π‘Ž RingHom 𝑦))
4140reseq2d 5981 . . . . . . . 8 (π‘₯ = π‘Ž β†’ ( I β†Ύ (π‘₯ RingHom 𝑦)) = ( I β†Ύ (π‘Ž RingHom 𝑦)))
42 oveq2 7419 . . . . . . . . 9 (𝑦 = 𝑏 β†’ (π‘Ž RingHom 𝑦) = (π‘Ž RingHom 𝑏))
4342reseq2d 5981 . . . . . . . 8 (𝑦 = 𝑏 β†’ ( I β†Ύ (π‘Ž RingHom 𝑦)) = ( I β†Ύ (π‘Ž RingHom 𝑏)))
4441, 43cbvmpov 7506 . . . . . . 7 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))) = (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RingHom 𝑏)))
4544a1i 11 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))) = (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RingHom 𝑏))))
4634a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘…))
47 eqidd 2733 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
48 fveq2 6891 . . . . . . . . . . . . 13 (𝑦 = 𝑏 β†’ (Baseβ€˜π‘¦) = (Baseβ€˜π‘))
49 fveq2 6891 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘Ž))
5048, 49oveqan12rd 7431 . . . . . . . . . . . 12 ((π‘₯ = π‘Ž ∧ 𝑦 = 𝑏) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
5150reseq2d 5981 . . . . . . . . . . 11 ((π‘₯ = π‘Ž ∧ 𝑦 = 𝑏) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
5251adantl 482 . . . . . . . . . 10 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ (π‘₯ = π‘Ž ∧ 𝑦 = 𝑏)) β†’ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
5334, 31eqsstrid 4030 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 βŠ† π‘ˆ)
5453sseld 3981 . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘Ž ∈ 𝐡 β†’ π‘Ž ∈ π‘ˆ))
5554com12 32 . . . . . . . . . . . 12 (π‘Ž ∈ 𝐡 β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
5655adantr 481 . . . . . . . . . . 11 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
5756impcom 408 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
5853sseld 3981 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ π‘ˆ))
5958adantld 491 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ π‘ˆ))
6059imp 407 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
61 ovexd 7446 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ∈ V)
6261resiexd 7220 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) ∈ V)
6347, 52, 57, 60, 62ovmpod 7562 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) = ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
6463reseq1d 5980 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏)) = (( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏)))
655adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
66 simprl 769 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ 𝐡)
67 simprr 771 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ 𝐡)
6813, 34, 65, 18, 66, 67ringchom 46996 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜π‘…)𝑏) = (π‘Ž RingHom 𝑏))
6968reseq2d 5981 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏)) = (( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†Ύ (π‘Ž RingHom 𝑏)))
70 eqid 2732 . . . . . . . . . . . 12 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
71 eqid 2732 . . . . . . . . . . . 12 (Baseβ€˜π‘) = (Baseβ€˜π‘)
7270, 71rhmf 20267 . . . . . . . . . . 11 (𝑓 ∈ (π‘Ž RingHom 𝑏) β†’ 𝑓:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘))
73 fvex 6904 . . . . . . . . . . . . . 14 (Baseβ€˜π‘) ∈ V
74 fvex 6904 . . . . . . . . . . . . . 14 (Baseβ€˜π‘Ž) ∈ V
7573, 74pm3.2i 471 . . . . . . . . . . . . 13 ((Baseβ€˜π‘) ∈ V ∧ (Baseβ€˜π‘Ž) ∈ V)
7675a1i 11 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((Baseβ€˜π‘) ∈ V ∧ (Baseβ€˜π‘Ž) ∈ V))
77 elmapg 8835 . . . . . . . . . . . 12 (((Baseβ€˜π‘) ∈ V ∧ (Baseβ€˜π‘Ž) ∈ V) β†’ (𝑓 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ↔ 𝑓:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)))
7876, 77syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑓 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ↔ 𝑓:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)))
7972, 78imbitrrid 245 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑓 ∈ (π‘Ž RingHom 𝑏) β†’ 𝑓 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
8079ssrdv 3988 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž RingHom 𝑏) βŠ† ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
8180resabs1d 6012 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†Ύ (π‘Ž RingHom 𝑏)) = ( I β†Ύ (π‘Ž RingHom 𝑏)))
8264, 69, 813eqtrrd 2777 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ (π‘Ž RingHom 𝑏)) = ((π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏)))
8335, 46, 82mpoeq123dva 7485 . . . . . 6 (πœ‘ β†’ (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RingHom 𝑏))) = (π‘Ž ∈ (Baseβ€˜π‘…), 𝑏 ∈ (Baseβ€˜π‘…) ↦ ((π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏))))
8439, 45, 833eqtrrd 2777 . . . . 5 (πœ‘ β†’ (π‘Ž ∈ (Baseβ€˜π‘…), 𝑏 ∈ (Baseβ€˜π‘…) ↦ ((π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏))) = 𝐺)
8538, 84opeq12d 4881 . . . 4 (πœ‘ β†’ ⟨((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) β†Ύ (Baseβ€˜π‘…)), (π‘Ž ∈ (Baseβ€˜π‘…), 𝑏 ∈ (Baseβ€˜π‘…) ↦ ((π‘Ž(π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))𝑏) β†Ύ (π‘Ž(Hom β€˜π‘…)𝑏)))⟩ = ⟨𝐹, 𝐺⟩)
8629, 85eqtr2d 2773 . . 3 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ = (⟨(π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)), (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))⟩ β†Ύf (Hom β€˜π‘…)))
8713, 5, 15, 19ringcval 46991 . . . 4 (πœ‘ β†’ 𝑅 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)))
8887oveq1d 7426 . . 3 (πœ‘ β†’ (𝑅 Func 𝑆) = (((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)) Func 𝑆))
8921, 86, 883eltr4d 2848 . 2 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 5149 . 2 (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
9189, 90sylibr 233 1 (πœ‘ β†’ 𝐹(𝑅 Func 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947  βŸ¨cop 4634   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413   ↑m cmap 8822  WUnicwun 10697  Basecbs 17146  Hom chom 17210   β†Ύcat cresc 17757   Func cfunc 17806   β†Ύf cresf 17809  SetCatcsetc 18027  ExtStrCatcestrc 18075  Ringcrg 20058   RingHom crh 20252  RingCatcringc 46986
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-wun 10699  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-fz 13487  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-hom 17223  df-cco 17224  df-0g 17389  df-cat 17614  df-cid 17615  df-homf 17616  df-ssc 17759  df-resc 17760  df-subc 17761  df-func 17810  df-resf 17813  df-setc 18028  df-estrc 18076  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-grp 18824  df-ghm 19092  df-mgp 19990  df-ur 20007  df-ring 20060  df-rnghom 20255  df-ringc 46988
This theorem is referenced by: (None)
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