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Theorem dffun9 5122
Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
dffun9  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  e.  ran  A  x A y ) )
Distinct variable group:    x, y, A

Proof of Theorem dffun9
StepHypRef Expression
1 dffun7 5120 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 vex 2663 . . . . . . . 8  |-  x  e. 
_V
3 vex 2663 . . . . . . . 8  |-  y  e. 
_V
42, 3brelrn 4742 . . . . . . 7  |-  ( x A y  ->  y  e.  ran  A )
54pm4.71ri 389 . . . . . 6  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
65mobii 2014 . . . . 5  |-  ( E* y  x A y  <->  E* y ( y  e. 
ran  A  /\  x A y ) )
7 df-rmo 2401 . . . . 5  |-  ( E* y  e.  ran  A  x A y  <->  E* y
( y  e.  ran  A  /\  x A y ) )
86, 7bitr4i 186 . . . 4  |-  ( E* y  x A y  <->  E* y  e.  ran  A  x A y )
98ralbii 2418 . . 3  |-  ( A. x  e.  dom  A E* y  x A y  <->  A. x  e.  dom  A E* y  e.  ran  A  x A y )
109anbi2i 452 . 2  |-  ( ( Rel  A  /\  A. x  e.  dom  A E* y  x A y )  <-> 
( Rel  A  /\  A. x  e.  dom  A E* y  e.  ran  A  x A y ) )
111, 10bitri 183 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  e.  ran  A  x A y ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1465   E*wmo 1978   A.wral 2393   E*wrmo 2396   class class class wbr 3899   dom cdm 4509   ran crn 4510   Rel wrel 4514   Fun wfun 5087
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rmo 2401  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095
This theorem is referenced by: (None)
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