Description: The inverse of the identity is the identity. Example 3.13 of Adamek p. 28. (Contributed by AV, 9-Apr-2020)
Ref | Expression | ||
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Hypotheses | invid.b | |- B = ( Base ` C ) |
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invid.i | |- I = ( Id ` C ) |
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invid.c | |- ( ph -> C e. Cat ) |
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invid.x | |- ( ph -> X e. B ) |
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Assertion | idinv | |- ( ph -> ( ( X ( Inv ` C ) X ) ` ( I ` X ) ) = ( I ` X ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invid.b | |- B = ( Base ` C ) |
|
2 | invid.i | |- I = ( Id ` C ) |
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3 | invid.c | |- ( ph -> C e. Cat ) |
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4 | invid.x | |- ( ph -> X e. B ) |
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5 | eqid | |- ( Inv ` C ) = ( Inv ` C ) |
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6 | 1 5 3 4 4 | invfun | |- ( ph -> Fun ( X ( Inv ` C ) X ) ) |
7 | 1 2 3 4 | invid | |- ( ph -> ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) ) |
8 | funbrfv | |- ( Fun ( X ( Inv ` C ) X ) -> ( ( I ` X ) ( X ( Inv ` C ) X ) ( I ` X ) -> ( ( X ( Inv ` C ) X ) ` ( I ` X ) ) = ( I ` X ) ) ) |
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9 | 6 7 8 | sylc | |- ( ph -> ( ( X ( Inv ` C ) X ) ` ( I ` X ) ) = ( I ` X ) ) |