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Metamath Proof Explorer

Theorem joindef

Description: Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018)

Ref Expression
Hypotheses joindef.u 
|- U = ( lub ` K )
joindef.j 
|- .\/ = ( join ` K )
joindef.k 
|- ( ph -> K e. V )
joindef.x 
|- ( ph -> X e. W )
joindef.y 
|- ( ph -> Y e. Z )
Assertion joindef 
|- ( ph -> ( <. X , Y >. e. dom .\/ <-> { X , Y } e. dom U ) )

Proof

Step Hyp Ref Expression
1 joindef.u 
 |-  U = ( lub ` K )
2 joindef.j 
 |-  .\/ = ( join ` K )
3 joindef.k 
 |-  ( ph -> K e. V )
4 joindef.x 
 |-  ( ph -> X e. W )
5 joindef.y 
 |-  ( ph -> Y e. Z )
6 1 2 joindm 
 |-  ( K e. V -> dom .\/ = { <. x , y >. | { x , y } e. dom U } )
7 6 eleq2d 
 |-  ( K e. V -> ( <. X , Y >. e. dom .\/ <-> <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } ) )
8 3 7 syl 
 |-  ( ph -> ( <. X , Y >. e. dom .\/ <-> <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } ) )
9 preq1 
 |-  ( x = X -> { x , y } = { X , y } )
10 9 eleq1d 
 |-  ( x = X -> ( { x , y } e. dom U <-> { X , y } e. dom U ) )
11 preq2 
 |-  ( y = Y -> { X , y } = { X , Y } )
12 11 eleq1d 
 |-  ( y = Y -> ( { X , y } e. dom U <-> { X , Y } e. dom U ) )
13 10 12 opelopabg 
 |-  ( ( X e. W /\ Y e. Z ) -> ( <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } <-> { X , Y } e. dom U ) )
14 4 5 13 syl2anc 
 |-  ( ph -> ( <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } <-> { X , Y } e. dom U ) )
15 8 14 bitrd 
 |-  ( ph -> ( <. X , Y >. e. dom .\/ <-> { X , Y } e. dom U ) )