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Theorem fun2cnv 5275
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 5271 . 2  |-  ( Fun  `' `' A  <->  A. x E* y 
y `' A x )
2 vex 2740 . . . . 5  |-  y  e. 
_V
3 vex 2740 . . . . 5  |-  x  e. 
_V
42, 3brcnv 4805 . . . 4  |-  ( y `' A x  <->  x A
y )
54mobii 2063 . . 3  |-  ( E* y  y `' A x 
<->  E* y  x A y )
65albii 1470 . 2  |-  ( A. x E* y  y `' A x  <->  A. x E* y  x A
y )
71, 6bitri 184 1  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wal 1351   E*wmo 2027   class class class wbr 4000   `'ccnv 4621   Fun wfun 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-fun 5213
This theorem is referenced by:  svrelfun  5276  fun11  5278
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