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Theorem fun2cnv 4983
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 4979 . 2  |-  ( Fun  `' `' A  <->  A. x E* y 
y `' A x )
2 vex 2604 . . . . 5  |-  y  e. 
_V
3 vex 2604 . . . . 5  |-  x  e. 
_V
42, 3brcnv 4536 . . . 4  |-  ( y `' A x  <->  x A
y )
54mobii 1978 . . 3  |-  ( E* y  y `' A x 
<->  E* y  x A y )
65albii 1399 . 2  |-  ( A. x E* y  y `' A x  <->  A. x E* y  x A
y )
71, 6bitri 182 1  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1282   E*wmo 1942   class class class wbr 3785   `'ccnv 4362   Fun wfun 4916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964
This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-fun 4924
This theorem is referenced by:  svrelfun  4984  fun11  4986
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