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Theorem dffun7 5150
Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 5151 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
Assertion
Ref Expression
dffun7  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
Distinct variable group:    x, y, A

Proof of Theorem dffun7
StepHypRef Expression
1 dffun6 5137 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 moabs 2048 . . . . . 6  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E* y  x A y ) )
3 vex 2689 . . . . . . . 8  |-  x  e. 
_V
43eldm 4736 . . . . . . 7  |-  ( x  e.  dom  A  <->  E. y  x A y )
54imbi1i 237 . . . . . 6  |-  ( ( x  e.  dom  A  ->  E* y  x A y )  <->  ( E. y  x A y  ->  E* y  x A
y ) )
62, 5bitr4i 186 . . . . 5  |-  ( E* y  x A y  <-> 
( x  e.  dom  A  ->  E* y  x A y ) )
76albii 1446 . . . 4  |-  ( A. x E* y  x A y  <->  A. x ( x  e.  dom  A  ->  E* y  x A
y ) )
8 df-ral 2421 . . . 4  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x
( x  e.  dom  A  ->  E* y  x A y ) )
97, 8bitr4i 186 . . 3  |-  ( A. x E* y  x A y  <->  A. x  e.  dom  A E* y  x A y )
109anbi2i 452 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. x  e. 
dom  A E* y  x A y ) )
111, 10bitri 183 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1329   E.wex 1468    e. wcel 1480   E*wmo 2000   A.wral 2416   class class class wbr 3929   dom cdm 4539   Rel wrel 4544   Fun wfun 5117
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-cnv 4547  df-co 4548  df-dm 4549  df-fun 5125
This theorem is referenced by:  dffun8  5151  dffun9  5152  funco  5163  funimaexglem  5206  frecuzrdgtcl  10192  frecuzrdgfunlem  10199
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