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Theorem joinfval 14121
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 2796 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 joinfval.j . . 3  |-  .\/  =  ( join `  K )
3 fveq2 5525 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 joinfval.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2333 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5525 . . . . . . 7  |-  ( p  =  K  ->  ( lub `  p )  =  ( lub `  K
) )
7 joinfval.u . . . . . . 7  |-  U  =  ( lub `  K
)
86, 7syl6eqr 2333 . . . . . 6  |-  ( p  =  K  ->  ( lub `  p )  =  U )
98fveq1d 5527 . . . . 5  |-  ( p  =  K  ->  (
( lub `  p
) `  { x ,  y } )  =  ( U `  { x ,  y } ) )
105, 5, 9mpt2eq123dv 5910 . . . 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 14110 . . . 4  |-  join  =  ( p  e.  _V  |->  ( x  e.  ( Base `  p ) ,  y  e.  ( Base `  p )  |->  ( ( lub `  p ) `
 { x ,  y } ) ) )
12 fvex 5539 . . . . . 6  |-  ( Base `  K )  e.  _V
134, 12eqeltri 2353 . . . . 5  |-  B  e. 
_V
1413, 13mpt2ex 6198 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( U `
 { x ,  y } ) )  e.  _V
1510, 11, 14fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  ( join `  K )  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
162, 15syl5eq 2327 . 2  |-  ( K  e.  _V  ->  .\/  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
171, 16syl 15 1  |-  ( K  e.  A  ->  .\/  =  ( x  e.  B ,  y  e.  B  |->  ( U `  {
x ,  y } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   {cpr 3641   ` cfv 5255    e. cmpt2 5860   Basecbs 13148   lubclub 14076   joincjn 14078
This theorem is referenced by:  joinval  14122  join0  14242  odumeet  14244  odujoin  14246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-join 14110
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