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Theorem mpt20 6386
Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
mpt20  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)

Proof of Theorem mpt20
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt2 6045 . 2  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) }
2 df-oprab 6044 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) }  =  {
w  |  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) ) }
3 noel 3592 . . . . . . 7  |-  -.  x  e.  (/)
4 simprll 739 . . . . . . 7  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )  ->  x  e.  (/) )
53, 4mto 169 . . . . . 6  |-  -.  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
65nex 1561 . . . . 5  |-  -.  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
76nex 1561 . . . 4  |-  -.  E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
87nex 1561 . . 3  |-  -.  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  (
( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
98abf 3621 . 2  |-  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) ) }  =  (/)
101, 2, 93eqtri 2428 1  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   (/)c0 3588   <.cop 3777   {coprab 6041    e. cmpt2 6042
This theorem is referenced by:  homffval  13872  comfffval  13879  natfval  14098  coafval  14174  xpchomfval  14231  xpccofval  14234  meet0  14519  join0  14520  plusffval  14657  grpsubfval  14802  oppglsm  15231  dvrfval  15744  scaffval  15923  psrmulr  16403  ipffval  16834  pcofval  18988  mendplusgfval  27361  mendmulrfval  27363  mendvscafval  27366
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589  df-oprab 6044  df-mpt2 6045
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