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Theorem funcringcsetc 47023
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 2731 . . . . . 6 (ExtStrCatβ€˜π‘ˆ) = (ExtStrCatβ€˜π‘ˆ)
2 funcringcsetc.s . . . . . 6 𝑆 = (SetCatβ€˜π‘ˆ)
3 eqid 2731 . . . . . 6 (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))
4 eqid 2731 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
5 funcringcsetc.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ WUni)
61, 5estrcbas 18081 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)))
76mpteq1d 5244 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ (Baseβ€˜π‘₯)))
8 mpoeq12 7485 . . . . . . 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 583 . . . . . 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 18106 . . . . 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 5150 . . . . 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 2731 . . . . . . 7 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1513, 14, 5ringcbas 46999 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘…) = (π‘ˆ ∩ Ring))
16 incom 4202 . . . . . 6 (π‘ˆ ∩ Ring) = (Ring ∩ π‘ˆ)
1715, 16eqtrdi 2787 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘…) = (Ring ∩ π‘ˆ))
18 eqid 2731 . . . . . 6 (Hom β€˜π‘…) = (Hom β€˜π‘…)
1913, 14, 5, 18ringchomfval 47000 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) = ( RingHom β†Ύ ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…))))
201, 5, 17, 19rhmsubcsetc 47011 . . . 4 (πœ‘ β†’ (Hom β€˜π‘…) ∈ (Subcatβ€˜(ExtStrCatβ€˜π‘ˆ)))
2112, 20funcres 17851 . . 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 7226 . . . . . 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 6905 . . . . . 6 (Hom β€˜π‘…) ∈ V
2524a1i 11 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) ∈ V)
26 mpoexga 8067 . . . . . 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 583 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) ∈ V)
2815, 19rhmresfn 46997 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) Fn ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…)))
2923, 25, 27, 28resfval2 17848 . . . 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 4229 . . . . . . . 8 (π‘ˆ ∩ Ring) βŠ† π‘ˆ
3115, 30eqsstrdi 4037 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) βŠ† π‘ˆ)
3231resmptd 6041 . . . . . 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 5244 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)))
3733, 36eqtr2d 2772 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)) = 𝐹)
3832, 37eqtrd 2771 . . . . 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 7419 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ (π‘₯ RingHom 𝑦) = (π‘Ž RingHom 𝑦))
4140reseq2d 5982 . . . . . . . 8 (π‘₯ = π‘Ž β†’ ( I β†Ύ (π‘₯ RingHom 𝑦)) = ( I β†Ύ (π‘Ž RingHom 𝑦)))
42 oveq2 7420 . . . . . . . . 9 (𝑦 = 𝑏 β†’ (π‘Ž RingHom 𝑦) = (π‘Ž RingHom 𝑏))
4342reseq2d 5982 . . . . . . . 8 (𝑦 = 𝑏 β†’ ( I β†Ύ (π‘Ž RingHom 𝑦)) = ( I β†Ύ (π‘Ž RingHom 𝑏)))
4441, 43cbvmpov 7507 . . . . . . 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 2732 . . . . . . . . . 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 6892 . . . . . . . . . . . . 13 (𝑦 = 𝑏 β†’ (Baseβ€˜π‘¦) = (Baseβ€˜π‘))
49 fveq2 6892 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘Ž))
5048, 49oveqan12rd 7432 . . . . . . . . . . . 12 ((π‘₯ = π‘Ž ∧ 𝑦 = 𝑏) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
5150reseq2d 5982 . . . . . . . . . . 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 481 . . . . . . . . . 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 4031 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 βŠ† π‘ˆ)
5453sseld 3982 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
5756impcom 407 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
5853sseld 3982 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ π‘ˆ))
5958adantld 490 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ π‘ˆ))
6059imp 406 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
61 ovexd 7447 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ∈ V)
6261resiexd 7221 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) ∈ V)
6347, 52, 57, 60, 62ovmpod 7563 . . . . . . . . 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 5981 . . . . . . . 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 480 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
66 simprl 768 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ 𝐡)
67 simprr 770 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ 𝐡)
6813, 34, 65, 18, 66, 67ringchom 47001 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜π‘…)𝑏) = (π‘Ž RingHom 𝑏))
6968reseq2d 5982 . . . . . . . 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 2731 . . . . . . . . . . . 12 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
71 eqid 2731 . . . . . . . . . . . 12 (Baseβ€˜π‘) = (Baseβ€˜π‘)
7270, 71rhmf 20377 . . . . . . . . . . 11 (𝑓 ∈ (π‘Ž RingHom 𝑏) β†’ 𝑓:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘))
73 fvex 6905 . . . . . . . . . . . . . 14 (Baseβ€˜π‘) ∈ V
74 fvex 6905 . . . . . . . . . . . . . 14 (Baseβ€˜π‘Ž) ∈ V
7573, 74pm3.2i 470 . . . . . . . . . . . . 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 8836 . . . . . . . . . . . 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 3989 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž RingHom 𝑏) βŠ† ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
8180resabs1d 6013 . . . . . . . 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 2776 . . . . . . 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 7486 . . . . . 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 2776 . . . . 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 4882 . . . 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 2772 . . 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 46996 . . . 4 (πœ‘ β†’ 𝑅 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)))
8887oveq1d 7427 . . 3 (πœ‘ β†’ (𝑅 Func 𝑆) = (((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)) Func 𝑆))
8921, 86, 883eltr4d 2847 . 2 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 5150 . 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 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∩ cin 3948  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232   I cid 5574   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414   ↑m cmap 8823  WUnicwun 10698  Basecbs 17149  Hom chom 17213   β†Ύcat cresc 17760   Func cfunc 17809   β†Ύf cresf 17812  SetCatcsetc 18030  ExtStrCatcestrc 18078  Ringcrg 20128   RingHom crh 20361  RingCatcringc 46991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-wun 10700  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-fz 13490  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-hom 17226  df-cco 17227  df-0g 17392  df-cat 17617  df-cid 17618  df-homf 17619  df-ssc 17762  df-resc 17763  df-subc 17764  df-func 17813  df-resf 17816  df-setc 18031  df-estrc 18079  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-grp 18859  df-ghm 19129  df-mgp 20030  df-ur 20077  df-ring 20130  df-rhm 20364  df-ringc 46993
This theorem is referenced by: (None)
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