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Theorem mpo0 5807
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
mpo0  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)

Proof of Theorem mpo0
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpo 5745 . 2  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) }
2 df-oprab 5744 . 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 3335 . . . . . . 7  |-  -.  x  e.  (/)
4 simprll 509 . . . . . . 7  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )  ->  x  e.  (/) )
53, 4mto 634 . . . . . 6  |-  -.  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ( ( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
65nex 1459 . . . . 5  |-  -.  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
76nex 1459 . . . 4  |-  -.  E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) )
87nex 1459 . . 3  |-  -.  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  (
( x  e.  (/)  /\  y  e.  B )  /\  z  =  C ) )
98abf 3374 . 2  |-  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\  ( ( x  e.  (/)  /\  y  e.  B
)  /\  z  =  C ) ) }  =  (/)
101, 2, 93eqtri 2140 1  |-  ( x  e.  (/) ,  y  e.  B  |->  C )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   (/)c0 3331   <.cop 3498   {coprab 5741    e. cmpo 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-oprab 5744  df-mpo 5745
This theorem is referenced by: (None)
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