Description: The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015)
Ref | Expression | ||
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Hypotheses | pi1val.g | |- G = ( J pi1 Y ) |
|
pi1val.1 | |- ( ph -> J e. ( TopOn ` X ) ) |
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pi1val.2 | |- ( ph -> Y e. X ) |
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pi1bas2.b | |- ( ph -> B = ( Base ` G ) ) |
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Assertion | pi1bas2 | |- ( ph -> B = ( U. B /. ( ~=ph ` J ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1val.g | |- G = ( J pi1 Y ) |
|
2 | pi1val.1 | |- ( ph -> J e. ( TopOn ` X ) ) |
|
3 | pi1val.2 | |- ( ph -> Y e. X ) |
|
4 | pi1bas2.b | |- ( ph -> B = ( Base ` G ) ) |
|
5 | eqid | |- ( J Om1 Y ) = ( J Om1 Y ) |
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6 | eqidd | |- ( ph -> ( Base ` ( J Om1 Y ) ) = ( Base ` ( J Om1 Y ) ) ) |
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7 | 1 2 3 5 4 6 | pi1buni | |- ( ph -> U. B = ( Base ` ( J Om1 Y ) ) ) |
8 | 1 2 3 5 4 7 | pi1bas | |- ( ph -> B = ( U. B /. ( ~=ph ` J ) ) ) |