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Theorem funmpt 4968
Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
Assertion
Ref Expression
funmpt  |-  Fun  (
x  e.  A  |->  B )

Proof of Theorem funmpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funopab4 4967 . 2  |-  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
2 df-mpt 3849 . . 3  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
32funeqi 4952 . 2  |-  ( Fun  ( x  e.  A  |->  B )  <->  Fun  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) } )
41, 3mpbir 144 1  |-  Fun  (
x  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   {copab 3846    |-> cmpt 3847   Fun wfun 4926
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-fun 4934
This theorem is referenced by:  funmpt2  4969  fmptco  5362  resfunexg  5414  mptexg  5418  brtpos2  5900  tposfun  5909  rdgtfr  6023  rdgruledefgg  6024  rdgon  6035  freccllem  6051  frecfcllem  6053  sizeinf  9802  sizeennn  9804  negfi  10248
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