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Metamath Proof Explorer

Theorem kqfeq

Description: Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015)

Ref Expression
Hypothesis kqval.2 
|- F = ( x e. X |-> { y e. J | x e. y } )
Assertion kqfeq 
|- ( ( J e. V /\ A e. X /\ B e. X ) -> ( ( F ` A ) = ( F ` B ) <-> A. y e. J ( A e. y <-> B e. y ) ) )

Proof

Step Hyp Ref Expression
1 kqval.2 
 |-  F = ( x e. X |-> { y e. J | x e. y } )
2 1 kqfval 
 |-  ( ( J e. V /\ A e. X ) -> ( F ` A ) = { y e. J | A e. y } )
3 2 3adant3 
 |-  ( ( J e. V /\ A e. X /\ B e. X ) -> ( F ` A ) = { y e. J | A e. y } )
4 1 kqfval 
 |-  ( ( J e. V /\ B e. X ) -> ( F ` B ) = { y e. J | B e. y } )
5 4 3adant2 
 |-  ( ( J e. V /\ A e. X /\ B e. X ) -> ( F ` B ) = { y e. J | B e. y } )
6 3 5 eqeq12d 
 |-  ( ( J e. V /\ A e. X /\ B e. X ) -> ( ( F ` A ) = ( F ` B ) <-> { y e. J | A e. y } = { y e. J | B e. y } ) )
7 rabbi 
 |-  ( A. y e. J ( A e. y <-> B e. y ) <-> { y e. J | A e. y } = { y e. J | B e. y } )
8 6 7 bitr4di 
 |-  ( ( J e. V /\ A e. X /\ B e. X ) -> ( ( F ` A ) = ( F ` B ) <-> A. y e. J ( A e. y <-> B e. y ) ) )