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Theorem mptrel 4737
Description: The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
mptrel  |-  Rel  (
x  e.  A  |->  B )

Proof of Theorem mptrel
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4050 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
21relopabi 4735 1  |-  Rel  (
x  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348    e. wcel 2141    |-> cmpt 4048   Rel wrel 4614
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-mpt 4050  df-xp 4615  df-rel 4616
This theorem is referenced by:  cnprcl2k  12959  psmetrel  13075  metrel  13095  xmetrel  13096  xmetf  13103  mopnrel  13194
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