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Theorem funcnv2 5391
Description: A simpler equivalence for single-rooted (see funcnv 5392). (Contributed by NM, 9-Aug-2004.)
Assertion
Ref Expression
funcnv2  |-  ( Fun  `' A  <->  A. y E* x  x A y )
Distinct variable group:    x, y, A

Proof of Theorem funcnv2
StepHypRef Expression
1 relcnv 5133 . . 3  |-  Rel  `' A
2 dffun6 5352 . . 3  |-  ( Fun  `' A  <->  ( Rel  `' A  /\  A. y E* x  y `' A x ) )
31, 2mpbiran 884 . 2  |-  ( Fun  `' A  <->  A. y E* x  y `' A x )
4 vex 2867 . . . . 5  |-  y  e. 
_V
5 vex 2867 . . . . 5  |-  x  e. 
_V
64, 5brcnv 4946 . . . 4  |-  ( y `' A x  <->  x A
y )
76mobii 2245 . . 3  |-  ( E* x  y `' A x 
<->  E* x  x A y )
87albii 1566 . 2  |-  ( A. y E* x  y `' A x  <->  A. y E* x  x A
y )
93, 8bitri 240 1  |-  ( Fun  `' A  <->  A. y E* x  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1540   E*wmo 2210   class class class wbr 4104   `'ccnv 4770   Rel wrel 4776   Fun wfun 5331
This theorem is referenced by:  funcnv  5392  fun2cnv  5394  fun11  5397  dff12  5519  1stconst  6294  2ndconst  6295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-fun 5339
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