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Theorem dffun7 5247
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 5248 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 5237 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 moabs 2189 . . . . . 6  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E* y  x A y ) )
3 vex 2793 . . . . . . . 8  |-  x  e. 
_V
43eldm 4876 . . . . . . 7  |-  ( x  e.  dom  A  <->  E. y  x A y )
54imbi1i 317 . . . . . 6  |-  ( ( x  e.  dom  A  ->  E* y  x A y )  <->  ( E. y  x A y  ->  E* y  x A
y ) )
62, 5bitr4i 245 . . . . 5  |-  ( E* y  x A y  <-> 
( x  e.  dom  A  ->  E* y  x A y ) )
76albii 1554 . . . 4  |-  ( A. x E* y  x A y  <->  A. x ( x  e.  dom  A  ->  E* y  x A
y ) )
8 df-ral 2550 . . . 4  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x
( x  e.  dom  A  ->  E* y  x A y ) )
97, 8bitr4i 245 . . 3  |-  ( A. x E* y  x A y  <->  A. x  e.  dom  A E* y  x A y )
109anbi2i 677 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. x  e. 
dom  A E* y  x A y ) )
111, 10bitri 242 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    e. wcel 1685   E*wmo 2146   A.wral 2545   class class class wbr 4025   dom cdm 4689   Rel wrel 4694   Fun wfun 5216
This theorem is referenced by:  dffun8  5248  dffun9  5249  brdom5  8150  imasaddfnlem  13425  imasvscafn  13434  funressnfv  27371
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-id 4309  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5224
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