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Theorem mpofun 5977
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 2197 . . . . . 6  |-  ( ( z  =  C  /\  w  =  C )  ->  z  =  w )
21ad2ant2l 508 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w )
32gen2 1450 . . . 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 2184 . . . . . 6  |-  ( z  =  w  ->  (
z  =  C  <->  w  =  C ) )
54anbi2d 464 . . . . 5  |-  ( z  =  w  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  w  =  C
) ) )
65mo4 2087 . . . 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 146 . . 3  |-  E* z
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )
87funoprab 5975 . 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 5880 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
119, 10eqtri 2198 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
1211funeqi 5238 . 2  |-  ( Fun 
F  <->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) } )
138, 12mpbir 146 1  |-  Fun  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351    = wceq 1353   E*wmo 2027    e. wcel 2148   Fun wfun 5211   {coprab 5876    e. cmpo 5877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-fun 5219  df-oprab 5879  df-mpo 5880
This theorem is referenced by:  elmpocl  6069  ofexg  6087  mpoexxg  6211  mpoexw  6214  mpoxopn0yelv  6240
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