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Theorem funcnv 5653
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 5652 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 3034 . . . . . . 7  |-  x  e. 
_V
2 vex 3034 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 5071 . . . . . 6  |-  ( x A y  ->  y  e.  ran  A )
43pm4.71ri 645 . . . . 5  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
54mobii 2342 . . . 4  |-  ( E* x  x A y  <->  E* x ( y  e. 
ran  A  /\  x A y ) )
6 moanimv 2380 . . . 4  |-  ( E* x ( y  e. 
ran  A  /\  x A y )  <->  ( y  e.  ran  A  ->  E* x  x A y ) )
75, 6bitri 257 . . 3  |-  ( E* x  x A y  <-> 
( y  e.  ran  A  ->  E* x  x A y ) )
87albii 1699 . 2  |-  ( A. y E* x  x A y  <->  A. y ( y  e.  ran  A  ->  E* x  x A
y ) )
9 funcnv2 5652 . 2  |-  ( Fun  `' A  <->  A. y E* x  x A y )
10 df-ral 2761 . 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 285 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    e. wcel 1904   E*wmo 2320   A.wral 2756   class class class wbr 4395   `'ccnv 4838   ran crn 4840   Fun wfun 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-fun 5591
This theorem is referenced by:  funcnv3  5654  fncnv  5657
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