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Theorem fun2cnv 5312
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 5309 . 2  |-  ( Fun  `' `' A  <->  A. x E* y 
y `' A x )
2 vex 2791 . . . . 5  |-  y  e. 
_V
3 vex 2791 . . . . 5  |-  x  e. 
_V
42, 3brcnv 4864 . . . 4  |-  ( y `' A x  <->  x A
y )
54mobii 2179 . . 3  |-  ( E* y  y `' A x 
<->  E* y  x A y )
65albii 1553 . 2  |-  ( A. x E* y  y `' A x  <->  A. x E* y  x A
y )
71, 6bitri 240 1  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1527   E*wmo 2144   class class class wbr 4023   `'ccnv 4688   Fun wfun 5249
This theorem is referenced by:  svrelfun  5313  fun11  5315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257
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