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Theorem dffun7 5553
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 5554 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 5542 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 moabs 2297 . . . . . 6  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E* y  x A y ) )
3 vex 3081 . . . . . . . 8  |-  x  e. 
_V
43eldm 5146 . . . . . . 7  |-  ( x  e.  dom  A  <->  E. y  x A y )
54imbi1i 325 . . . . . 6  |-  ( ( x  e.  dom  A  ->  E* y  x A y )  <->  ( E. y  x A y  ->  E* y  x A
y ) )
62, 5bitr4i 252 . . . . 5  |-  ( E* y  x A y  <-> 
( x  e.  dom  A  ->  E* y  x A y ) )
76albii 1611 . . . 4  |-  ( A. x E* y  x A y  <->  A. x ( x  e.  dom  A  ->  E* y  x A
y ) )
8 df-ral 2804 . . . 4  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x
( x  e.  dom  A  ->  E* y  x A y ) )
97, 8bitr4i 252 . . 3  |-  ( A. x E* y  x A y  <->  A. x  e.  dom  A E* y  x A y )
109anbi2i 694 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. x  e. 
dom  A E* y  x A y ) )
111, 10bitri 249 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368   E.wex 1587    e. wcel 1758   E*wmo 2263   A.wral 2799   class class class wbr 4401   dom cdm 4949   Rel wrel 4954   Fun wfun 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-cnv 4957  df-co 4958  df-dm 4959  df-fun 5529
This theorem is referenced by:  dffun8  5554  dffun9  5555  brdom5  8808  imasaddfnlem  14586  imasvscafn  14595  funressnfv  30183
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