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Theorem mpt20 5718
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 5657 . 2  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) }
2 df-oprab 5656 . 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 3290 . . . . . . 7  |-  -.  x  e.  (/)
4 simprll 504 . . . . . . 7  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )  ->  x  e.  (/) )
53, 4mto 623 . . . . . 6  |-  -.  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
65nex 1434 . . . . 5  |-  -.  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
76nex 1434 . . . 4  |-  -.  E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
87nex 1434 . . 3  |-  -.  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  (
( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
98abf 3326 . 2  |-  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) ) }  =  (/)
101, 2, 93eqtri 2112 1  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   (/)c0 3286   <.cop 3449   {coprab 5653    |-> cmpt2 5654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-oprab 5656  df-mpt2 5657
This theorem is referenced by: (None)
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