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Theorem mpt20 6199
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 5863 . 2  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) }
2 df-oprab 5862 . 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 3459 . . . . . . 7  |-  -.  x  e.  (/)
4 simprll 738 . . . . . . 7  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )  ->  x  e.  (/) )
53, 4mto 167 . . . . . 6  |-  -.  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
65nex 1542 . . . . 5  |-  -.  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
76nex 1542 . . . 4  |-  -.  E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
87nex 1542 . . 3  |-  -.  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  (
( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
98abf 3488 . 2  |-  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) ) }  =  (/)
101, 2, 93eqtri 2307 1  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   (/)c0 3455   <.cop 3643   {coprab 5859    e. cmpt2 5860
This theorem is referenced by:  homffval  13594  comfffval  13601  natfval  13820  coafval  13896  xpchomfval  13953  xpccofval  13956  meet0  14241  join0  14242  plusffval  14379  grpsubfval  14524  oppglsm  14953  dvrfval  15466  scaffval  15645  psrmulr  16129  ipffval  16552  pcofval  18508  mendplusgfval  27493  mendmulrfval  27495  mendvscafval  27498
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-oprab 5862  df-mpt2 5863
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