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Theorem svrelfun 5278
Description: A single-valued relation is a function. (See fun2cnv 5277 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
) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem svrelfun
StepHypRef Expression
1 dffun6 5236 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 fun2cnv 5277 . . 3  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
32anbi2i 677 . 2  |-  ( ( Rel  A  /\  Fun  `' `' A )  <->  ( Rel  A  /\  A. x E* y  x A y ) )
41, 3bitr4i 245 1  |-  ( Fun 
A  <->  ( Rel  A  /\  Fun  `' `' A
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   A.wal 1528   E*wmo 2145   class class class wbr 4024   `'ccnv 4687   Rel wrel 4693   Fun wfun 5215
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-fun 5223
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