Description: A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | setcbas.c | |- C = ( SetCat ` U ) |
|
setcbas.u | |- ( ph -> U e. V ) |
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setchomfval.h | |- H = ( Hom ` C ) |
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setchom.x | |- ( ph -> X e. U ) |
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setchom.y | |- ( ph -> Y e. U ) |
||
Assertion | elsetchom | |- ( ph -> ( F e. ( X H Y ) <-> F : X --> Y ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setcbas.c | |- C = ( SetCat ` U ) |
|
2 | setcbas.u | |- ( ph -> U e. V ) |
|
3 | setchomfval.h | |- H = ( Hom ` C ) |
|
4 | setchom.x | |- ( ph -> X e. U ) |
|
5 | setchom.y | |- ( ph -> Y e. U ) |
|
6 | 1 2 3 4 5 | setchom | |- ( ph -> ( X H Y ) = ( Y ^m X ) ) |
7 | 6 | eleq2d | |- ( ph -> ( F e. ( X H Y ) <-> F e. ( Y ^m X ) ) ) |
8 | 5 4 | elmapd | |- ( ph -> ( F e. ( Y ^m X ) <-> F : X --> Y ) ) |
9 | 7 8 | bitrd | |- ( ph -> ( F e. ( X H Y ) <-> F : X --> Y ) ) |