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Metamath Proof Explorer

Theorem isfne4b

Description: A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009) (Revised by Mario Carneiro, 11-Sep-2015)

Ref Expression
Hypotheses isfne.1 
|- X = U. A
isfne.2 
|- Y = U. B
Assertion isfne4b 
|- ( B e. V -> ( A Fne B <-> ( X = Y /\ ( topGen ` A ) C_ ( topGen ` B ) ) ) )

Proof

Step Hyp Ref Expression
1 isfne.1 
 |-  X = U. A
2 isfne.2 
 |-  Y = U. B
3 1 2 isfne4 
 |-  ( A Fne B <-> ( X = Y /\ A C_ ( topGen ` B ) ) )
4 simpr 
 |-  ( ( B e. V /\ X = Y ) -> X = Y )
5 4 1 2 3eqtr3g 
 |-  ( ( B e. V /\ X = Y ) -> U. A = U. B )
6 uniexg 
 |-  ( B e. V -> U. B e. _V )
7 6 adantr 
 |-  ( ( B e. V /\ X = Y ) -> U. B e. _V )
8 5 7 eqeltrd 
 |-  ( ( B e. V /\ X = Y ) -> U. A e. _V )
9 uniexb 
 |-  ( A e. _V <-> U. A e. _V )
10 8 9 sylibr 
 |-  ( ( B e. V /\ X = Y ) -> A e. _V )
11 simpl 
 |-  ( ( B e. V /\ X = Y ) -> B e. V )
12 tgss3 
 |-  ( ( A e. _V /\ B e. V ) -> ( ( topGen ` A ) C_ ( topGen ` B ) <-> A C_ ( topGen ` B ) ) )
13 10 11 12 syl2anc 
 |-  ( ( B e. V /\ X = Y ) -> ( ( topGen ` A ) C_ ( topGen ` B ) <-> A C_ ( topGen ` B ) ) )
14 13 pm5.32da 
 |-  ( B e. V -> ( ( X = Y /\ ( topGen ` A ) C_ ( topGen ` B ) ) <-> ( X = Y /\ A C_ ( topGen ` B ) ) ) )
15 3 14 bitr4id 
 |-  ( B e. V -> ( A Fne B <-> ( X = Y /\ ( topGen ` A ) C_ ( topGen ` B ) ) ) )