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Theorem dffun9 5282
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 5280 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 vex 2791 . . . . . . . 8  |-  x  e. 
_V
3 vex 2791 . . . . . . . 8  |-  y  e. 
_V
42, 3brelrn 4909 . . . . . . 7  |-  ( x A y  ->  y  e.  ran  A )
54pm4.71ri 614 . . . . . 6  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
65mobii 2179 . . . . 5  |-  ( E* y  x A y  <->  E* y ( y  e. 
ran  A  /\  x A y ) )
7 df-rmo 2551 . . . . 5  |-  ( E* y  e.  ran  A  x A y  <->  E* y
( y  e.  ran  A  /\  x A y ) )
86, 7bitr4i 243 . . . 4  |-  ( E* y  x A y  <->  E* y  e.  ran  A  x A y )
98ralbii 2567 . . 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 675 . 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 240 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:    <-> wb 176    /\ wa 358    e. wcel 1684   E*wmo 2144   A.wral 2543   E*wrmo 2546   class class class wbr 4023   dom cdm 4689   ran crn 4690   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  brdom4  8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rmo 2551  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257
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