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Theorem joinfval 15500
Description: Value of join function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove joinfval2 15501 first to reduce net proof size (existence part)?
Hypotheses
Ref Expression
joinfval.u  |-  U  =  ( lub `  K
)
joinfval.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
joinfval  |-  ( K  e.  V  ->  .\/  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } U z } )
Distinct variable groups:    x, y,
z, K    z, U
Allowed substitution hints:    U( x, y)    .\/ ( x, y, z)    V( x, y, z)

Proof of Theorem joinfval
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 3102 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 joinfval.j . . 3  |-  .\/  =  ( join `  K )
3 fvex 5862 . . . . . . 7  |-  ( Base `  K )  e.  _V
4 moeq 3259 . . . . . . . 8  |-  E* z 
z  =  ( U `
 { x ,  y } )
54a1i 11 . . . . . . 7  |-  ( ( x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  ->  E* z  z  =  ( U `  { x ,  y } ) )
6 eqid 2441 . . . . . . 7  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) )  /\  z  =  ( U `  { x ,  y } ) ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) }
73, 3, 5, 6oprabex 6769 . . . . . 6  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) )  /\  z  =  ( U `  { x ,  y } ) ) }  e.  _V
87a1i 11 . . . . 5  |-  ( K  e.  _V  ->  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) )  /\  z  =  ( U `  { x ,  y } ) ) }  e.  _V )
9 joinfval.u . . . . . . . . . . . 12  |-  U  =  ( lub `  K
)
109lubfun 15479 . . . . . . . . . . 11  |-  Fun  U
11 funbrfv2b 5898 . . . . . . . . . . 11  |-  ( Fun 
U  ->  ( {
x ,  y } U z  <->  ( {
x ,  y }  e.  dom  U  /\  ( U `  { x ,  y } )  =  z ) ) )
1210, 11ax-mp 5 . . . . . . . . . 10  |-  ( { x ,  y } U z  <->  ( {
x ,  y }  e.  dom  U  /\  ( U `  { x ,  y } )  =  z ) )
13 eqid 2441 . . . . . . . . . . . . . 14  |-  ( Base `  K )  =  (
Base `  K )
14 eqid 2441 . . . . . . . . . . . . . 14  |-  ( le
`  K )  =  ( le `  K
)
15 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( K  e.  _V  /\  { x ,  y }  e.  dom  U )  ->  K  e.  _V )
16 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( K  e.  _V  /\  { x ,  y }  e.  dom  U )  ->  { x ,  y }  e.  dom  U )
1713, 14, 9, 15, 16lubelss 15481 . . . . . . . . . . . . 13  |-  ( ( K  e.  _V  /\  { x ,  y }  e.  dom  U )  ->  { x ,  y }  C_  ( Base `  K ) )
1817ex 434 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  ( { x ,  y }  e.  dom  U  ->  { x ,  y }  C_  ( Base `  K ) ) )
19 vex 3096 . . . . . . . . . . . . 13  |-  x  e. 
_V
20 vex 3096 . . . . . . . . . . . . 13  |-  y  e. 
_V
2119, 20prss 4165 . . . . . . . . . . . 12  |-  ( ( x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  <->  { x ,  y }  C_  ( Base `  K )
)
2218, 21syl6ibr 227 . . . . . . . . . . 11  |-  ( K  e.  _V  ->  ( { x ,  y }  e.  dom  U  ->  ( x  e.  (
Base `  K )  /\  y  e.  ( Base `  K ) ) ) )
23 eqcom 2450 . . . . . . . . . . . . 13  |-  ( ( U `  { x ,  y } )  =  z  <->  z  =  ( U `  { x ,  y } ) )
2423biimpi 194 . . . . . . . . . . . 12  |-  ( ( U `  { x ,  y } )  =  z  ->  z  =  ( U `  { x ,  y } ) )
2524a1i 11 . . . . . . . . . . 11  |-  ( K  e.  _V  ->  (
( U `  {
x ,  y } )  =  z  -> 
z  =  ( U `
 { x ,  y } ) ) )
2622, 25anim12d 563 . . . . . . . . . 10  |-  ( K  e.  _V  ->  (
( { x ,  y }  e.  dom  U  /\  ( U `  { x ,  y } )  =  z )  ->  ( (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  /\  z  =  ( U `  { x ,  y } ) ) ) )
2712, 26syl5bi 217 . . . . . . . . 9  |-  ( K  e.  _V  ->  ( { x ,  y } U z  -> 
( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) ) )
2827alrimiv 1704 . . . . . . . 8  |-  ( K  e.  _V  ->  A. z
( { x ,  y } U z  ->  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) ) )
2928alrimiv 1704 . . . . . . 7  |-  ( K  e.  _V  ->  A. y A. z ( { x ,  y } U
z  ->  ( (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) )  /\  z  =  ( U `  { x ,  y } ) ) ) )
3029alrimiv 1704 . . . . . 6  |-  ( K  e.  _V  ->  A. x A. y A. z ( { x ,  y } U z  -> 
( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) ) )
31 ssoprab2 6334 . . . . . 6  |-  ( A. x A. y A. z
( { x ,  y } U z  ->  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) )  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y } U z }  C_  {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) } )
3230, 31syl 16 . . . . 5  |-  ( K  e.  _V  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y } U z }  C_  {
<. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( Base `  K
)  /\  y  e.  ( Base `  K )
)  /\  z  =  ( U `  { x ,  y } ) ) } )
338, 32ssexd 4580 . . . 4  |-  ( K  e.  _V  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y } U z }  e.  _V )
34 fveq2 5852 . . . . . . . 8  |-  ( p  =  K  ->  ( lub `  p )  =  ( lub `  K
) )
3534, 9syl6eqr 2500 . . . . . . 7  |-  ( p  =  K  ->  ( lub `  p )  =  U )
3635breqd 4444 . . . . . 6  |-  ( p  =  K  ->  ( { x ,  y }  ( lub `  p
) z  <->  { x ,  y } U
z ) )
3736oprabbidv 6332 . . . . 5  |-  ( p  =  K  ->  { <. <.
x ,  y >. ,  z >.  |  {
x ,  y }  ( lub `  p
) z }  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } U z } )
38 df-join 15475 . . . . 5  |-  join  =  ( p  e.  _V  |->  { <. <. x ,  y
>. ,  z >.  |  { x ,  y }  ( lub `  p
) z } )
3937, 38fvmptg 5935 . . . 4  |-  ( ( K  e.  _V  /\  {
<. <. x ,  y
>. ,  z >.  |  { x ,  y } U z }  e.  _V )  -> 
( join `  K )  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } U z } )
4033, 39mpdan 668 . . 3  |-  ( K  e.  _V  ->  ( join `  K )  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } U z } )
412, 40syl5eq 2494 . 2  |-  ( K  e.  _V  ->  .\/  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } U z } )
421, 41syl 16 1  |-  ( K  e.  V  ->  .\/  =  { <. <. x ,  y
>. ,  z >.  |  { x ,  y } U z } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1379    = wceq 1381    e. wcel 1802   E*wmo 2267   _Vcvv 3093    C_ wss 3458   {cpr 4012   class class class wbr 4433   dom cdm 4985   Fun wfun 5568   ` cfv 5574   {coprab 6278   Basecbs 14504   lecple 14576   lubclub 15440   joincjn 15442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-oprab 6281  df-lub 15473  df-join 15475
This theorem is referenced by:  joinfval2  15501  join0  15637  odumeet  15639  odujoin  15641
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