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Theorem dffun7 5138
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 5139 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 5128 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
2 moabs 2157 . . . . . 6  |-  ( E* y  x A y  <-> 
( E. y  x A y  ->  E* y  x A y ) )
3 vex 2730 . . . . . . . 8  |-  x  e. 
_V
43eldm 4783 . . . . . . 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 2513 . . . 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 678 . 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 1532   E.wex 1537    e. wcel 1621   E*wmo 2115   A.wral 2509   class class class wbr 3920   dom cdm 4580   Rel wrel 4585   Fun wfun 4586
This theorem is referenced by:  dffun8  5139  dffun9  5140  brdom5  8038  imasaddfnlem  13304  imasvscafn  13313
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-cnv 4596  df-co 4597  df-dm 4598  df-fun 4602
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