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Theorem fun2cnv 5182
Description: The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that  A is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
fun2cnv  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
Distinct variable group:    x, y, A

Proof of Theorem fun2cnv
StepHypRef Expression
1 funcnv2 5178 . 2  |-  ( Fun  `' `' A  <->  A. x E* y 
y `' A x )
2 vex 2684 . . . . 5  |-  y  e. 
_V
3 vex 2684 . . . . 5  |-  x  e. 
_V
42, 3brcnv 4717 . . . 4  |-  ( y `' A x  <->  x A
y )
54mobii 2034 . . 3  |-  ( E* y  y `' A x 
<->  E* y  x A y )
65albii 1446 . 2  |-  ( A. x E* y  y `' A x  <->  A. x E* y  x A
y )
71, 6bitri 183 1  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1329   E*wmo 1998   class class class wbr 3924   `'ccnv 4533   Fun wfun 5112
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-fun 5120
This theorem is referenced by:  svrelfun  5183  fun11  5185
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