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Metamath Proof Explorer

Theorem dminss

Description: An upper bound for intersection with a domain. Theorem 40 of Suppes p. 66, who calls it "somewhat surprising". (Contributed by NM, 11-Aug-2004)

Ref Expression
Assertion dminss 
|- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Proof

Step Hyp Ref Expression
1 19.8a 
 |-  ( ( x e. A /\ x R y ) -> E. x ( x e. A /\ x R y ) )
2 1 ancoms 
 |-  ( ( x R y /\ x e. A ) -> E. x ( x e. A /\ x R y ) )
3 vex 
 |-  y e. _V
4 3 elima2 
 |-  ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
5 2 4 sylibr 
 |-  ( ( x R y /\ x e. A ) -> y e. ( R " A ) )
6 simpl 
 |-  ( ( x R y /\ x e. A ) -> x R y )
7 vex 
 |-  x e. _V
8 3 7 brcnv 
 |-  ( y `' R x <-> x R y )
9 6 8 sylibr 
 |-  ( ( x R y /\ x e. A ) -> y `' R x )
10 5 9 jca 
 |-  ( ( x R y /\ x e. A ) -> ( y e. ( R " A ) /\ y `' R x ) )
11 10 eximi 
 |-  ( E. y ( x R y /\ x e. A ) -> E. y ( y e. ( R " A ) /\ y `' R x ) )
12 7 eldm 
 |-  ( x e. dom R <-> E. y x R y )
13 12 anbi1i 
 |-  ( ( x e. dom R /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
14 elin 
 |-  ( x e. ( dom R i^i A ) <-> ( x e. dom R /\ x e. A ) )
15 19.41v 
 |-  ( E. y ( x R y /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
16 13 14 15 3bitr4i 
 |-  ( x e. ( dom R i^i A ) <-> E. y ( x R y /\ x e. A ) )
17 7 elima2 
 |-  ( x e. ( `' R " ( R " A ) ) <-> E. y ( y e. ( R " A ) /\ y `' R x ) )
18 11 16 17 3imtr4i 
 |-  ( x e. ( dom R i^i A ) -> x e. ( `' R " ( R " A ) ) )
19 18 ssriv 
 |-  ( dom R i^i A ) C_ ( `' R " ( R " A ) )