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Theorem funcnv 5310
Description: The converse of a class is a function iff the class is single-rooted, which means that for any  y in the range of  A there is at most one  x such that  x A
y. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5309 for a simpler version. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
funcnv  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
Distinct variable group:    x, y, A

Proof of Theorem funcnv
StepHypRef Expression
1 vex 2791 . . . . . . 7  |-  x  e. 
_V
2 vex 2791 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 4909 . . . . . 6  |-  ( x A y  ->  y  e.  ran  A )
43pm4.71ri 614 . . . . 5  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
54mobii 2179 . . . 4  |-  ( E* x  x A y  <->  E* x ( y  e. 
ran  A  /\  x A y ) )
6 moanimv 2201 . . . 4  |-  ( E* x ( y  e. 
ran  A  /\  x A y )  <->  ( y  e.  ran  A  ->  E* x  x A y ) )
75, 6bitri 240 . . 3  |-  ( E* x  x A y  <-> 
( y  e.  ran  A  ->  E* x  x A y ) )
87albii 1553 . 2  |-  ( A. y E* x  x A y  <->  A. y ( y  e.  ran  A  ->  E* x  x A
y ) )
9 funcnv2 5309 . 2  |-  ( Fun  `' A  <->  A. y E* x  x A y )
10 df-ral 2548 . 2  |-  ( A. y  e.  ran  A E* x  x A y  <->  A. y
( y  e.  ran  A  ->  E* x  x A y ) )
118, 9, 103bitr4i 268 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684   E*wmo 2144   A.wral 2543   class class class wbr 4023   `'ccnv 4688   ran crn 4690   Fun wfun 5249
This theorem is referenced by:  funcnv3  5311  fncnv  5314
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-dm 4699  df-rn 4700  df-fun 5257
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