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Theorem svrelfun 5313
Description: A single-valued relation is a function. (See fun2cnv 5312 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 5270 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 fun2cnv 5312 . . 3  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
32anbi2i 675 . 2  |-  ( ( Rel  A  /\  Fun  `' `' A )  <->  ( Rel  A  /\  A. x E* y  x A y ) )
41, 3bitr4i 243 1  |-  ( Fun 
A  <->  ( Rel  A  /\  Fun  `' `' A
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1527   E*wmo 2144   class class class wbr 4023   `'ccnv 4688   Rel wrel 4694   Fun wfun 5249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257
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