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Theorem svrelfun 5281
Description: A single-valued relation is a function. (See fun2cnv 5280 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
svrelfun  |-  ( Fun 
A  <->  ( Rel  A  /\  Fun  `' `' A
) )

Proof of Theorem svrelfun
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun6 5230 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 fun2cnv 5280 . . 3  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
32anbi2i 457 . 2  |-  ( ( Rel  A  /\  Fun  `' `' A )  <->  ( Rel  A  /\  A. x E* y  x A y ) )
41, 3bitr4i 187 1  |-  ( Fun 
A  <->  ( Rel  A  /\  Fun  `' `' A
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105   A.wal 1351   E*wmo 2027   class class class wbr 4003   `'ccnv 4625   Rel wrel 4631   Fun wfun 5210
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-fun 5218
This theorem is referenced by: (None)
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