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Theorem joinfval 14436
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 2956 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 joinfval.j . . 3  |-  .\/  =  ( join `  K )
3 fveq2 5720 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 joinfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2485 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5720 . . . . . . 7  |-  ( p  =  K  ->  ( lub `  p )  =  ( lub `  K
) )
7 joinfval.u . . . . . . 7  |-  U  =  ( lub `  K
)
86, 7syl6eqr 2485 . . . . . 6  |-  ( p  =  K  ->  ( lub `  p )  =  U )
98fveq1d 5722 . . . . 5  |-  ( p  =  K  ->  (
( lub `  p
) `  { x ,  y } )  =  ( U `  { x ,  y } ) )
105, 5, 9mpt2eq123dv 6128 . . . 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 14425 . . . 4  |-  join  =  ( p  e.  _V  |->  ( x  e.  ( Base `  p ) ,  y  e.  ( Base `  p )  |->  ( ( lub `  p ) `
 { x ,  y } ) ) )
12 fvex 5734 . . . . . 6  |-  ( Base `  K )  e.  _V
134, 12eqeltri 2505 . . . . 5  |-  B  e. 
_V
1413, 13mpt2ex 6417 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( U `
 { x ,  y } ) )  e.  _V
1510, 11, 14fvmpt 5798 . . 3  |-  ( K  e.  _V  ->  ( join `  K )  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
162, 15syl5eq 2479 . 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 1652    e. wcel 1725   _Vcvv 2948   {cpr 3807   ` cfv 5446    e. cmpt2 6075   Basecbs 13461   lubclub 14391   joincjn 14393
This theorem is referenced by:  joinval  14437  join0  14557  odumeet  14559  odujoin  14561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-join 14425
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