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Theorem mpofun 5839
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpofun.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
mpofun  |-  Fun  F
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem mpofun
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2135 . . . . . 6  |-  ( ( z  =  C  /\  w  =  C )  ->  z  =  w )
21ad2ant2l 497 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w )
32gen2 1409 . . . 4  |-  A. z A. w ( ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w )
4 eqeq1 2122 . . . . . 6  |-  ( z  =  w  ->  (
z  =  C  <->  w  =  C ) )
54anbi2d 457 . . . . 5  |-  ( z  =  w  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  w  =  C
) ) )
65mo4 2036 . . . 4  |-  ( E* z ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  A. z A. w ( ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w ) )
73, 6mpbir 145 . . 3  |-  E* z
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )
87funoprab 5837 . 2  |-  Fun  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
9 mpofun.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
10 df-mpo 5745 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
119, 10eqtri 2136 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
1211funeqi 5112 . 2  |-  ( Fun 
F  <->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) } )
138, 12mpbir 145 1  |-  Fun  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314    e. wcel 1463   E*wmo 1976   Fun wfun 5085   {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-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-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-fun 5093  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  elmpocl  5934  ofexg  5952  mpoexxg  6074  mpoxopn0yelv  6102
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