MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcnv Unicode version

Theorem funcnv 5275
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 5274 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 2792 . . . . . . 7  |-  x  e. 
_V
2 vex 2792 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 4908 . . . . . 6  |-  ( x A y  ->  y  e.  ran  A )
43pm4.71ri 617 . . . . 5  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
54mobii 2180 . . . 4  |-  ( E* x  x A y  <->  E* x ( y  e. 
ran  A  /\  x A y ) )
6 moanimv 2202 . . . 4  |-  ( E* x ( y  e. 
ran  A  /\  x A y )  <->  ( y  e.  ran  A  ->  E* x  x A y ) )
75, 6bitri 242 . . 3  |-  ( E* x  x A y  <-> 
( y  e.  ran  A  ->  E* x  x A y ) )
87albii 1558 . 2  |-  ( A. y E* x  x A y  <->  A. y ( y  e.  ran  A  ->  E* x  x A
y ) )
9 funcnv2 5274 . 2  |-  ( Fun  `' A  <->  A. y E* x  x A y )
10 df-ral 2549 . 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 270 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    e. wcel 1688   E*wmo 2145   A.wral 2544   class class class wbr 4024   `'ccnv 4687   ran crn 4689   Fun wfun 5215
This theorem is referenced by:  funcnv3  5276  fncnv  5279
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-fun 5223
  Copyright terms: Public domain W3C validator