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Theorem fun2cnv 5169
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 5166 . 2  |-  ( Fun  `' `' A  <->  A. x E* y 
y `' A x )
2 vex 2730 . . . . 5  |-  y  e. 
_V
3 vex 2730 . . . . 5  |-  x  e. 
_V
42, 3brcnv 4771 . . . 4  |-  ( y `' A x  <->  x A
y )
54mobii 2149 . . 3  |-  ( E* y  y `' A x 
<->  E* y  x A y )
65albii 1554 . 2  |-  ( A. x E* y  y `' A x  <->  A. x E* y  x A
y )
71, 6bitri 242 1  |-  ( Fun  `' `' A  <->  A. x E* y  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1532   E*wmo 2115   class class class wbr 3920   `'ccnv 4579   Fun wfun 4586
This theorem is referenced by:  svrelfun  5170  fun11  5172
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-fun 4602
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