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Metamath Proof Explorer

Theorem kqtopon

Description: The Kolmogorov quotient is a topology on the quotient set. (Contributed by Mario Carneiro, 25-Aug-2015)

Ref Expression
Hypothesis kqval.2 
|- F = ( x e. X |-> { y e. J | x e. y } )
Assertion kqtopon 
|- ( J e. ( TopOn ` X ) -> ( KQ ` J ) e. ( TopOn ` ran F ) )

Proof

Step Hyp Ref Expression
1 kqval.2 
 |-  F = ( x e. X |-> { y e. J | x e. y } )
2 1 kqval 
 |-  ( J e. ( TopOn ` X ) -> ( KQ ` J ) = ( J qTop F ) )
3 1 kqffn 
 |-  ( J e. ( TopOn ` X ) -> F Fn X )
4 dffn4 
 |-  ( F Fn X <-> F : X -onto-> ran F )
5 3 4 sylib 
 |-  ( J e. ( TopOn ` X ) -> F : X -onto-> ran F )
6 qtoptopon 
 |-  ( ( J e. ( TopOn ` X ) /\ F : X -onto-> ran F ) -> ( J qTop F ) e. ( TopOn ` ran F ) )
7 5 6 mpdan 
 |-  ( J e. ( TopOn ` X ) -> ( J qTop F ) e. ( TopOn ` ran F ) )
8 2 7 eqeltrd 
 |-  ( J e. ( TopOn ` X ) -> ( KQ ` J ) e. ( TopOn ` ran F ) )