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Metamath Proof Explorer

Theorem uniqs

Description: The union of a quotient set, like uniqsw but with a weaker antecedent: only the restriction of R by A needs to be a set, not R itself, see e.g. cnvepima . (Contributed by NM, 9-Dec-2008) (Revised by Peter Mazsa, 20-Jun-2019)

Ref Expression
Assertion uniqs 
|- ( ( R |` A ) e. V -> U. ( A /. R ) = ( R " A ) )

Proof

Step Hyp Ref Expression
1 elecex 
 |-  ( ( R |` A ) e. V -> ( x e. A -> [ x ] R e. _V ) )
2 1 ralrimiv 
 |-  ( ( R |` A ) e. V -> A. x e. A [ x ] R e. _V )
3 dfiun2g 
 |-  ( A. x e. A [ x ] R e. _V -> U_ x e. A [ x ] R = U. { y | E. x e. A y = [ x ] R } )
4 2 3 syl 
 |-  ( ( R |` A ) e. V -> U_ x e. A [ x ] R = U. { y | E. x e. A y = [ x ] R } )
5 4 eqcomd 
 |-  ( ( R |` A ) e. V -> U. { y | E. x e. A y = [ x ] R } = U_ x e. A [ x ] R )
6 df-qs 
 |-  ( A /. R ) = { y | E. x e. A y = [ x ] R }
7 6 unieqi 
 |-  U. ( A /. R ) = U. { y | E. x e. A y = [ x ] R }
8 df-ec 
 |-  [ x ] R = ( R " { x } )
9 8 a1i 
 |-  ( x e. A -> [ x ] R = ( R " { x } ) )
10 9 iuneq2i 
 |-  U_ x e. A [ x ] R = U_ x e. A ( R " { x } )
11 imaiun 
 |-  ( R " U_ x e. A { x } ) = U_ x e. A ( R " { x } )
12 iunid 
 |-  U_ x e. A { x } = A
13 12 imaeq2i 
 |-  ( R " U_ x e. A { x } ) = ( R " A )
14 10 11 13 3eqtr2ri 
 |-  ( R " A ) = U_ x e. A [ x ] R
15 5 7 14 3eqtr4g 
 |-  ( ( R |` A ) e. V -> U. ( A /. R ) = ( R " A ) )