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Theorem funcnv 5259
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 5258 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 2733 . . . . . . 7  |-  x  e. 
_V
2 vex 2733 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 4844 . . . . . 6  |-  ( x A y  ->  y  e.  ran  A )
43pm4.71ri 390 . . . . 5  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
54mobii 2056 . . . 4  |-  ( E* x  x A y  <->  E* x ( y  e. 
ran  A  /\  x A y ) )
6 moanimv 2094 . . . 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 1463 . 2  |-  ( A. y E* x  x A y  <->  A. y ( y  e.  ran  A  ->  E* x  x A
y ) )
9 funcnv2 5258 . 2  |-  ( Fun  `' A  <->  A. y E* x  x A y )
10 df-ral 2453 . 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 1346   E*wmo 2020    e. wcel 2141   A.wral 2448   class class class wbr 3989   `'ccnv 4610   ran crn 4612   Fun wfun 5192
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200
This theorem is referenced by:  funcnv3  5260  fncnv  5264
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