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Theorem funcrngcsetc 46386
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 46387, using cofuval2 17781 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 46385, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18045. (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‘₯ RngHomo 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 2733 . . . . . 6 (ExtStrCatβ€˜π‘ˆ) = (ExtStrCatβ€˜π‘ˆ)
2 funcrngcsetc.s . . . . . 6 𝑆 = (SetCatβ€˜π‘ˆ)
3 eqid 2733 . . . . . 6 (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ))
4 eqid 2733 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
5 funcrngcsetc.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ WUni)
61, 5estrcbas 18020 . . . . . . 7 (πœ‘ β†’ π‘ˆ = (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)))
76mpteq1d 5204 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜(ExtStrCatβ€˜π‘ˆ)) ↦ (Baseβ€˜π‘₯)))
8 mpoeq12 7434 . . . . . . 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 585 . . . . . 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 18045 . . . . 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 5110 . . . . 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 2733 . . . . . . 7 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1513, 14, 5rngcbas 46353 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘…) = (π‘ˆ ∩ Rng))
16 incom 4165 . . . . . 6 (π‘ˆ ∩ Rng) = (Rng ∩ π‘ˆ)
1715, 16eqtrdi 2789 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘…) = (Rng ∩ π‘ˆ))
18 eqid 2733 . . . . . 6 (Hom β€˜π‘…) = (Hom β€˜π‘…)
1913, 14, 5, 18rngchomfval 46354 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) = ( RngHomo β†Ύ ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…))))
201, 5, 17, 19rnghmsubcsetc 46365 . . . 4 (πœ‘ β†’ (Hom β€˜π‘…) ∈ (Subcatβ€˜(ExtStrCatβ€˜π‘ˆ)))
2112, 20funcres 17790 . . 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 7175 . . . . . 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 6859 . . . . . 6 (Hom β€˜π‘…) ∈ V
2524a1i 11 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) ∈ V)
26 mpoexga 8014 . . . . . 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 585 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))) ∈ V)
2815, 19rnghmresfn 46351 . . . . 5 (πœ‘ β†’ (Hom β€˜π‘…) Fn ((Baseβ€˜π‘…) Γ— (Baseβ€˜π‘…)))
2923, 25, 27, 28resfval2 17787 . . . 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 4192 . . . . . . . 8 (π‘ˆ ∩ Rng) βŠ† π‘ˆ
3115, 30eqsstrdi 4002 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) βŠ† π‘ˆ)
3231resmptd 5998 . . . . . 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 5204 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)))
3733, 36eqtr2d 2774 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (Baseβ€˜π‘…) ↦ (Baseβ€˜π‘₯)) = 𝐹)
3832, 37eqtrd 2773 . . . . 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‘₯ RngHomo 𝑦))))
40 oveq1 7368 . . . . . . . . 9 (π‘₯ = π‘Ž β†’ (π‘₯ RngHomo 𝑦) = (π‘Ž RngHomo 𝑦))
4140reseq2d 5941 . . . . . . . 8 (π‘₯ = π‘Ž β†’ ( I β†Ύ (π‘₯ RngHomo 𝑦)) = ( I β†Ύ (π‘Ž RngHomo 𝑦)))
42 oveq2 7369 . . . . . . . . 9 (𝑦 = 𝑏 β†’ (π‘Ž RngHomo 𝑦) = (π‘Ž RngHomo 𝑏))
4342reseq2d 5941 . . . . . . . 8 (𝑦 = 𝑏 β†’ ( I β†Ύ (π‘Ž RngHomo 𝑦)) = ( I β†Ύ (π‘Ž RngHomo 𝑏)))
4441, 43cbvmpov 7456 . . . . . . 7 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHomo 𝑦))) = (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RngHomo 𝑏)))
4544a1i 11 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RngHomo 𝑦))) = (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RngHomo 𝑏))))
4634a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘…))
47 eqidd 2734 . . . . . . . . . 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 6846 . . . . . . . . . . . . 13 (𝑦 = 𝑏 β†’ (Baseβ€˜π‘¦) = (Baseβ€˜π‘))
49 fveq2 6846 . . . . . . . . . . . . 13 (π‘₯ = π‘Ž β†’ (Baseβ€˜π‘₯) = (Baseβ€˜π‘Ž))
5048, 49oveqan12rd 7381 . . . . . . . . . . . 12 ((π‘₯ = π‘Ž ∧ 𝑦 = 𝑏) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
5150reseq2d 5941 . . . . . . . . . . 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 483 . . . . . . . . . 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 3996 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐡 βŠ† π‘ˆ)
5453sseld 3947 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
5756impcom 409 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
5853sseld 3947 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑏 ∈ 𝐡 β†’ 𝑏 ∈ π‘ˆ))
5958adantld 492 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ 𝑏 ∈ π‘ˆ))
6059imp 408 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
61 ovexd 7396 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ∈ V)
6261resiexd 7170 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) ∈ V)
6347, 52, 57, 60, 62ovmpod 7511 . . . . . . . . 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 5940 . . . . . . . 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 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
66 simprl 770 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ 𝐡)
67 simprr 772 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ 𝐡)
6813, 34, 65, 18, 66, 67rngchom 46355 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜π‘…)𝑏) = (π‘Ž RngHomo 𝑏))
6968reseq2d 5941 . . . . . . . 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 RngHomo 𝑏)))
70 eqid 2733 . . . . . . . . . . . 12 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
71 eqid 2733 . . . . . . . . . . . 12 (Baseβ€˜π‘) = (Baseβ€˜π‘)
7270, 71rnghmf 46287 . . . . . . . . . . 11 (𝑓 ∈ (π‘Ž RngHomo 𝑏) β†’ 𝑓:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘))
73 fvex 6859 . . . . . . . . . . . . . 14 (Baseβ€˜π‘) ∈ V
74 fvex 6859 . . . . . . . . . . . . . 14 (Baseβ€˜π‘Ž) ∈ V
7573, 74pm3.2i 472 . . . . . . . . . . . . 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 8784 . . . . . . . . . . . 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, 78syl5ibr 246 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑓 ∈ (π‘Ž RngHomo 𝑏) β†’ 𝑓 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
8079ssrdv 3954 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž RngHomo 𝑏) βŠ† ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
8180resabs1d 5972 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (( I β†Ύ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†Ύ (π‘Ž RngHomo 𝑏)) = ( I β†Ύ (π‘Ž RngHomo 𝑏)))
8264, 69, 813eqtrrd 2778 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ( I β†Ύ (π‘Ž RngHomo 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 7435 . . . . . 6 (πœ‘ β†’ (π‘Ž ∈ 𝐡, 𝑏 ∈ 𝐡 ↦ ( I β†Ύ (π‘Ž RngHomo 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 2778 . . . . 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 4842 . . . 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 2774 . . 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 46350 . . . 4 (πœ‘ β†’ 𝑅 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)))
8887oveq1d 7376 . . 3 (πœ‘ β†’ (𝑅 Func 𝑆) = (((ExtStrCatβ€˜π‘ˆ) β†Ύcat (Hom β€˜π‘…)) Func 𝑆))
8921, 86, 883eltr4d 2849 . 2 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90 df-br 5110 . 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447   ∩ cin 3913  βŸ¨cop 4596   class class class wbr 5109   ↦ cmpt 5192   I cid 5534   β†Ύ cres 5639  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363   ↑m cmap 8771  WUnicwun 10644  Basecbs 17091  Hom chom 17152   β†Ύcat cresc 17699   Func cfunc 17748   β†Ύf cresf 17751  SetCatcsetc 17969  ExtStrCatcestrc 18017  Rngcrng 46262   RngHomo crngh 46273  RngCatcrngc 46345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-wun 10646  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-hom 17165  df-cco 17166  df-0g 17331  df-cat 17556  df-cid 17557  df-homf 17558  df-ssc 17701  df-resc 17702  df-subc 17703  df-func 17752  df-resf 17755  df-setc 17970  df-estrc 18018  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-mhm 18609  df-grp 18759  df-ghm 19014  df-abl 19573  df-mgp 19905  df-mgmhm 46163  df-rng 46263  df-rnghomo 46275  df-rngc 46347
This theorem is referenced by: (None)
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