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Metamath Proof Explorer

Theorem quseccl0

Description: Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015) Generalization of quseccl for arbitrary sets G . (Revised by AV, 24-Feb-2025)

Ref Expression
Hypotheses quseccl0.e 
|- .~ = ( G ~QG S )
quseccl0.h 
|- H = ( G /s .~ )
quseccl0.c 
|- C = ( Base ` G )
quseccl0.b 
|- B = ( Base ` H )
Assertion quseccl0 
|- ( ( G e. V /\ X e. C ) -> [ X ] .~ e. B )

Proof

Step Hyp Ref Expression
1 quseccl0.e 
 |-  .~ = ( G ~QG S )
2 quseccl0.h 
 |-  H = ( G /s .~ )
3 quseccl0.c 
 |-  C = ( Base ` G )
4 quseccl0.b 
 |-  B = ( Base ` H )
5 1 ovexi 
 |-  .~ e. _V
6 5 ecelqsi 
 |-  ( X e. C -> [ X ] .~ e. ( C /. .~ ) )
7 6 adantl 
 |-  ( ( G e. V /\ X e. C ) -> [ X ] .~ e. ( C /. .~ ) )
8 2 a1i 
 |-  ( ( G e. V /\ X e. C ) -> H = ( G /s .~ ) )
9 3 a1i 
 |-  ( ( G e. V /\ X e. C ) -> C = ( Base ` G ) )
10 5 a1i 
 |-  ( ( G e. V /\ X e. C ) -> .~ e. _V )
11 simpl 
 |-  ( ( G e. V /\ X e. C ) -> G e. V )
12 8 9 10 11 qusbas 
 |-  ( ( G e. V /\ X e. C ) -> ( C /. .~ ) = ( Base ` H ) )
13 12 4 eqtr4di 
 |-  ( ( G e. V /\ X e. C ) -> ( C /. .~ ) = B )
14 7 13 eleqtrd 
 |-  ( ( G e. V /\ X e. C ) -> [ X ] .~ e. B )