ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funcnv Unicode version

Theorem funcnv 5231
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 5230 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 2715 . . . . . . 7  |-  x  e. 
_V
2 vex 2715 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 4819 . . . . . 6  |-  ( x A y  ->  y  e.  ran  A )
43pm4.71ri 390 . . . . 5  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
54mobii 2043 . . . 4  |-  ( E* x  x A y  <->  E* x ( y  e. 
ran  A  /\  x A y ) )
6 moanimv 2081 . . . 4  |-  ( E* x ( y  e. 
ran  A  /\  x A y )  <->  ( y  e.  ran  A  ->  E* x  x A y ) )
75, 6bitri 183 . . 3  |-  ( E* x  x A y  <-> 
( y  e.  ran  A  ->  E* x  x A y ) )
87albii 1450 . 2  |-  ( A. y E* x  x A y  <->  A. y ( y  e.  ran  A  ->  E* x  x A
y ) )
9 funcnv2 5230 . 2  |-  ( Fun  `' A  <->  A. y E* x  x A y )
10 df-ral 2440 . 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 211 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1333   E*wmo 2007    e. wcel 2128   A.wral 2435   class class class wbr 3965   `'ccnv 4585   ran crn 4587   Fun wfun 5164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-fun 5172
This theorem is referenced by:  funcnv3  5232  fncnv  5236
  Copyright terms: Public domain W3C validator