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Metamath Proof Explorer

Theorem eqinfd

Description: Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020)

Ref Expression
Hypotheses infexd.1 
|- ( ph -> R Or A )
eqinfd.2 
|- ( ph -> C e. A )
eqinfd.3 
|- ( ( ph /\ y e. B ) -> -. y R C )
eqinfd.4 
|- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
Assertion eqinfd 
|- ( ph -> inf ( B , A , R ) = C )

Proof

Step Hyp Ref Expression
1 infexd.1 
 |-  ( ph -> R Or A )
2 eqinfd.2 
 |-  ( ph -> C e. A )
3 eqinfd.3 
 |-  ( ( ph /\ y e. B ) -> -. y R C )
4 eqinfd.4 
 |-  ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
5 3 ralrimiva 
 |-  ( ph -> A. y e. B -. y R C )
6 4 expr 
 |-  ( ( ph /\ y e. A ) -> ( C R y -> E. z e. B z R y ) )
7 6 ralrimiva 
 |-  ( ph -> A. y e. A ( C R y -> E. z e. B z R y ) )
8 1 eqinf 
 |-  ( ph -> ( ( C e. A /\ A. y e. B -. y R C /\ A. y e. A ( C R y -> E. z e. B z R y ) ) -> inf ( B , A , R ) = C ) )
9 2 5 7 8 mp3and 
 |-  ( ph -> inf ( B , A , R ) = C )