Metamath Proof Explorer

Theorem issconn

Description: The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015)

Ref Expression
Assertion issconn 
|- ( J e. SConn <-> ( J e. PConn /\ A. f e. ( II Cn J ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) ) )

Proof

Step Hyp Ref Expression
1 oveq2 
 |-  ( j = J -> ( II Cn j ) = ( II Cn J ) )
2 fveq2 
 |-  ( j = J -> ( ~=ph ` j ) = ( ~=ph ` J ) )
3 2 breqd 
 |-  ( j = J -> ( f ( ~=ph ` j ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) <-> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) )
4 3 imbi2d 
 |-  ( j = J -> ( ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` j ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) <-> ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) ) )
5 1 4 raleqbidv 
 |-  ( j = J -> ( A. f e. ( II Cn j ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` j ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) <-> A. f e. ( II Cn J ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) ) )
6 df-sconn 
 |-  SConn = { j e. PConn | A. f e. ( II Cn j ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` j ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) }
7 5 6 elrab2 
 |-  ( J e. SConn <-> ( J e. PConn /\ A. f e. ( II Cn J ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` J ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) ) )