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Metamath Proof Explorer

Theorem marypha2lem1

Description: Lemma for marypha2 . Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015)

Ref Expression
Hypothesis marypha2lem.t 
|- T = U_ x e. A ( { x } X. ( F ` x ) )
Assertion marypha2lem1 
|- T C_ ( A X. U. ran F )

Proof

Step Hyp Ref Expression
1 marypha2lem.t 
 |-  T = U_ x e. A ( { x } X. ( F ` x ) )
2 iunss 
 |-  ( U_ x e. A ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) <-> A. x e. A ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) )
3 snssi 
 |-  ( x e. A -> { x } C_ A )
4 fvssunirn 
 |-  ( F ` x ) C_ U. ran F
5 xpss12 
 |-  ( ( { x } C_ A /\ ( F ` x ) C_ U. ran F ) -> ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) )
6 3 4 5 sylancl 
 |-  ( x e. A -> ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) )
7 2 6 mprgbir 
 |-  U_ x e. A ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F )
8 1 7 eqsstri 
 |-  T C_ ( A X. U. ran F )