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Theorem joinfval 14372
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
joinfval.b  |-  B  =  ( Base `  K
)
joinfval.u  |-  U  =  ( lub `  K
)
joinfval.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
joinfval  |-  ( K  e.  A  ->  .\/  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
Distinct variable groups:    x, y, B    x, K, y
Allowed substitution hints:    A( x, y)    U( x, y)    .\/ ( x, y)

Proof of Theorem joinfval
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 2908 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 joinfval.j . . 3  |-  .\/  =  ( join `  K )
3 fveq2 5669 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 joinfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2438 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5669 . . . . . . 7  |-  ( p  =  K  ->  ( lub `  p )  =  ( lub `  K
) )
7 joinfval.u . . . . . . 7  |-  U  =  ( lub `  K
)
86, 7syl6eqr 2438 . . . . . 6  |-  ( p  =  K  ->  ( lub `  p )  =  U )
98fveq1d 5671 . . . . 5  |-  ( p  =  K  ->  (
( lub `  p
) `  { x ,  y } )  =  ( U `  { x ,  y } ) )
105, 5, 9mpt2eq123dv 6076 . . . 4  |-  ( p  =  K  ->  (
x  e.  ( Base `  p ) ,  y  e.  ( Base `  p
)  |->  ( ( lub `  p ) `  {
x ,  y } ) )  =  ( x  e.  B , 
y  e.  B  |->  ( U `  { x ,  y } ) ) )
11 df-join 14361 . . . 4  |-  join  =  ( p  e.  _V  |->  ( x  e.  ( Base `  p ) ,  y  e.  ( Base `  p )  |->  ( ( lub `  p ) `
 { x ,  y } ) ) )
12 fvex 5683 . . . . . 6  |-  ( Base `  K )  e.  _V
134, 12eqeltri 2458 . . . . 5  |-  B  e. 
_V
1413, 13mpt2ex 6365 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( U `
 { x ,  y } ) )  e.  _V
1510, 11, 14fvmpt 5746 . . 3  |-  ( K  e.  _V  ->  ( join `  K )  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
162, 15syl5eq 2432 . 2  |-  ( K  e.  _V  ->  .\/  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
171, 16syl 16 1  |-  ( K  e.  A  ->  .\/  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2900   {cpr 3759   ` cfv 5395    e. cmpt2 6023   Basecbs 13397   lubclub 14327   joincjn 14329
This theorem is referenced by:  joinval  14373  join0  14493  odumeet  14495  odujoin  14497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-join 14361
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