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Metamath Proof Explorer

Theorem restuni6

Description: The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses restuni6.1 
|- ( ph -> A e. V )
restuni6.2 
|- ( ph -> B e. W )
Assertion restuni6 
|- ( ph -> U. ( A |`t B ) = ( U. A i^i B ) )

Proof

Step Hyp Ref Expression
1 restuni6.1 
 |-  ( ph -> A e. V )
2 restuni6.2 
 |-  ( ph -> B e. W )
3 eqid 
 |-  U. A = U. A
4 3 restin 
 |-  ( ( A e. V /\ B e. W ) -> ( A |`t B ) = ( A |`t ( B i^i U. A ) ) )
5 1 2 4 syl2anc 
 |-  ( ph -> ( A |`t B ) = ( A |`t ( B i^i U. A ) ) )
6 5 unieqd 
 |-  ( ph -> U. ( A |`t B ) = U. ( A |`t ( B i^i U. A ) ) )
7 inss2 
 |-  ( B i^i U. A ) C_ U. A
8 7 a1i 
 |-  ( ph -> ( B i^i U. A ) C_ U. A )
9 1 8 restuni4 
 |-  ( ph -> U. ( A |`t ( B i^i U. A ) ) = ( B i^i U. A ) )
10 incom 
 |-  ( B i^i U. A ) = ( U. A i^i B )
11 10 a1i 
 |-  ( ph -> ( B i^i U. A ) = ( U. A i^i B ) )
12 6 9 11 3eqtrd 
 |-  ( ph -> U. ( A |`t B ) = ( U. A i^i B ) )