Description: According to df-subc , the subcategories ( SubcatC ) of a category C are subsets of the homomorphisms of C (see subcssc and subcss2 ). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020)
Ref | Expression | ||
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Hypotheses | drhmsubc.c | |- C = ( U i^i DivRing ) |
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drhmsubc.j | |- J = ( r e. C , s e. C |-> ( r RingHom s ) ) |
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Assertion | drhmsubc | |- ( U e. V -> J e. ( Subcat ` ( RingCat ` U ) ) ) |
Step | Hyp | Ref | Expression |
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1 | drhmsubc.c | |- C = ( U i^i DivRing ) |
|
2 | drhmsubc.j | |- J = ( r e. C , s e. C |-> ( r RingHom s ) ) |
|
3 | drngring | |- ( r e. DivRing -> r e. Ring ) |
|
4 | 3 | rgen | |- A. r e. DivRing r e. Ring |
5 | 4 1 2 | srhmsubc | |- ( U e. V -> J e. ( Subcat ` ( RingCat ` U ) ) ) |