Metamath Proof Explorer


Theorem pi1bas2

Description: The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015)

Ref Expression
Hypotheses pi1val.g
|- G = ( J pi1 Y )
pi1val.1
|- ( ph -> J e. ( TopOn ` X ) )
pi1val.2
|- ( ph -> Y e. X )
pi1bas2.b
|- ( ph -> B = ( Base ` G ) )
Assertion pi1bas2
|- ( ph -> B = ( U. B /. ( ~=ph ` J ) ) )

Proof

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 )
6 eqidd
 |-  ( ph -> ( Base ` ( J Om1 Y ) ) = ( Base ` ( J Om1 Y ) ) )
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 ) ) )