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Theorem svrelfun 5505
Description: A single-valued relation is a function. (See fun2cnv 5504 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 5460 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 fun2cnv 5504 . . 3  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
32anbi2i 676 . 2  |-  ( ( Rel  A  /\  Fun  `' `' A )  <->  ( Rel  A  /\  A. x E* y  x A y ) )
41, 3bitr4i 244 1  |-  ( Fun 
A  <->  ( Rel  A  /\  Fun  `' `' A
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   A.wal 1549   E*wmo 2281   class class class wbr 4204   `'ccnv 4868   Rel wrel 4874   Fun wfun 5439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-fun 5447
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