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Theorem funcrngcsetc 20575
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 20576, using cofuval2 17870 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 20574, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18137. (Contributed by AV, 26-Mar-2020.)
Hypotheses
Ref Expression
funcrngcsetc.r 𝑅 = (RngCatβ€˜π‘ˆ)
funcrngcsetc.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcrngcsetc.b 𝐡 = (Baseβ€˜π‘…)
funcrngcsetc.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcrngcsetc.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcrngcsetc.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHom 𝑦))))
Assertion
Ref Expression
funcrngcsetc (πœ‘ β†’ 𝐹(𝑅 Func 𝑆)𝐺)
Distinct variable groups:   π‘₯,𝐡,𝑦   π‘₯,𝑅,𝑦   π‘₯,𝑆   π‘₯,π‘ˆ,𝑦   πœ‘,π‘₯,𝑦
Allowed substitution hints:   𝑆(𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem funcrngcsetc
Dummy variables π‘Ž 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . . . 6 (ExtStrCatβ€˜π‘ˆ) = (ExtStrCatβ€˜π‘ˆ)
2 funcrngcsetc.s . . . . . 6 𝑆 = (SetCatβ€˜π‘ˆ)
3 eqid 2725 . . . . . 6 (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))
4 eqid 2725 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
5 funcrngcsetc.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ WUni)
61, 5estrcbas 18112 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)))
76mpteq1d 5238 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ (Baseβ€˜π‘₯)))
8 mpoeq12 7489 . . . . . . 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 582 . . . . . 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 18137 . . . . 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 5144 . . . . 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 funcrngcsetc.r . . . . . . 7 𝑅 = (RngCatβ€˜π‘ˆ)
14 eqid 2725 . . . . . . 7 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1513, 14, 5rngcbas 20556 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘…) = (π‘ˆ ∩ Rng))
16 incom 4195 . . . . . 6 (π‘ˆ ∩ Rng) = (Rng ∩ π‘ˆ)
1715, 16eqtrdi 2781 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘…) = (Rng ∩ π‘ˆ))
18 eqid 2725 . . . . . 6 (Hom β€˜π‘…) = (Hom β€˜π‘…)
1913, 14, 5, 18rngchomfval 20557 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) = ( RngHom β†Ύ ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…))))
201, 5, 17, 19rnghmsubcsetc 20568 . . . 4 (πœ‘ β†’ (Hom β€˜π‘…) ∈ (Subcatβ€˜(ExtStrCatβ€˜π‘ˆ)))
2112, 20funcres 17879 . . 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 7228 . . . . . 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 8078 . . . . . 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 582 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) ∈ V)
2815, 19rnghmresfn 20554 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) Fn ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…)))
2923, 25, 27, 28resfval2 17876 . . . 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 4223 . . . . . . . 8 (π‘ˆ ∩ Rng) βŠ† π‘ˆ
3115, 30eqsstrdi 4027 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) βŠ† π‘ˆ)
3231resmptd 6039 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) β†Ύ (Baseβ€˜π‘…)) = (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)))
33 funcrngcsetc.f . . . . . . 7 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
34 funcrngcsetc.b . . . . . . . . 9 𝐡 = (Baseβ€˜π‘…)
3534a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))
3635mpteq1d 5238 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)))
3733, 36eqtr2d 2766 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)) = 𝐹)
3832, 37eqtrd 2765 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) β†Ύ (Baseβ€˜π‘…)) = 𝐹)
39 funcrngcsetc.g . . . . . 6 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHom 𝑦))))
40 oveq1 7422 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ (π‘₯ RngHom 𝑦) = (π‘Ž RngHom 𝑦))
4140reseq2d 5979 . . . . . . . 8 (π‘₯ = π‘Ž β†’ ( I β†Ύ (π‘₯ RngHom 𝑦)) = ( I β†Ύ (π‘Ž RngHom 𝑦)))
42 oveq2 7423 . . . . . . . . 9 (𝑦 = 𝑏 β†’ (π‘Ž RngHom 𝑦) = (π‘Ž RngHom 𝑏))
4342reseq2d 5979 . . . . . . . 8 (𝑦 = 𝑏 β†’ ( I β†Ύ (π‘Ž RngHom 𝑦)) = ( I β†Ύ (π‘Ž RngHom 𝑏)))
4441, 43cbvmpov 7511 . . . . . . 7 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHom 𝑦))) = (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RngHom 𝑏)))
4544a1i 11 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHom 𝑦))) = (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RngHom 𝑏))))
4634a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘…))
47 eqidd 2726 . . . . . . . . . 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 7435 . . . . . . . . . . . 12 ((π‘₯ = π‘Ž ∧ 𝑦 = 𝑏) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
5150reseq2d 5979 . . . . . . . . . . 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 480 . . . . . . . . . 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 4021 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 βŠ† π‘ˆ)
5453sseld 3971 . . . . . . . . . . . . 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 479 . . . . . . . . . . 11 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
5756impcom 406 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
5853sseld 3971 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ π‘ˆ))
5958adantld 489 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ π‘ˆ))
6059imp 405 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
61 ovexd 7450 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ∈ V)
6261resiexd 7223 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) ∈ V)
6347, 52, 57, 60, 62ovmpod 7569 . . . . . . . . 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 5978 . . . . . . . 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 479 . . . . . . . . . 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, 67rngchom 20558 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜π‘…)𝑏) = (π‘Ž RngHom 𝑏))
6968reseq2d 5979 . . . . . . . 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 RngHom 𝑏)))
70 eqid 2725 . . . . . . . . . . . 12 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
71 eqid 2725 . . . . . . . . . . . 12 (Baseβ€˜π‘) = (Baseβ€˜π‘)
7270, 71rnghmf 20389 . . . . . . . . . . 11 (𝑓 ∈ (π‘Ž RngHom 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 469 . . . . . . . . . . . . 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 8854 . . . . . . . . . . . 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 RngHom 𝑏) β†’ 𝑓 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
8079ssrdv 3978 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž RngHom 𝑏) βŠ† ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
8180resabs1d 6007 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†Ύ (π‘Ž RngHom 𝑏)) = ( I β†Ύ (π‘Ž RngHom 𝑏)))
8264, 69, 813eqtrrd 2770 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ (π‘Ž RngHom 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 7490 . . . . . 6 (πœ‘ β†’ (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RngHom 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 2770 . . . . 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 4877 . . . 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 2766 . . 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, 19rngcval 20553 . . . 4 (πœ‘ β†’ 𝑅 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)))
8887oveq1d 7430 . . 3 (πœ‘ β†’ (𝑅 Func 𝑆) = (((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)) Func 𝑆))
8921, 86, 883eltr4d 2840 . 2 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 5144 . 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∩ cin 3939  βŸ¨cop 4630   class class class wbr 5143   ↦ cmpt 5226   I cid 5569   β†Ύ cres 5674  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417   ↑m cmap 8841  WUnicwun 10721  Basecbs 17177  Hom chom 17241   β†Ύcat cresc 17788   Func cfunc 17837   β†Ύf cresf 17840  SetCatcsetc 18061  ExtStrCatcestrc 18109  Rngcrng 20094   RngHom crnghm 20375  RngCatcrngc 20551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-pm 8844  df-ixp 8913  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-wun 10723  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-uz 12851  df-fz 13515  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-hom 17254  df-cco 17255  df-0g 17420  df-cat 17645  df-cid 17646  df-homf 17647  df-ssc 17790  df-resc 17791  df-subc 17792  df-func 17841  df-resf 17844  df-setc 18062  df-estrc 18110  df-mgm 18597  df-mgmhm 18649  df-sgrp 18676  df-mnd 18692  df-mhm 18737  df-grp 18895  df-ghm 19170  df-abl 19740  df-mgp 20077  df-rng 20095  df-rnghm 20377  df-rngc 20552
This theorem is referenced by: (None)
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