MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffun9 Structured version   Unicode version

Theorem dffun9 5473
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 5471 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 vex 2951 . . . . . . . 8  |-  x  e. 
_V
3 vex 2951 . . . . . . . 8  |-  y  e. 
_V
42, 3brelrn 5092 . . . . . . 7  |-  ( x A y  ->  y  e.  ran  A )
54pm4.71ri 615 . . . . . 6  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
65mobii 2316 . . . . 5  |-  ( E* y  x A y  <->  E* y ( y  e. 
ran  A  /\  x A y ) )
7 df-rmo 2705 . . . . 5  |-  ( E* y  e.  ran  A  x A y  <->  E* y
( y  e.  ran  A  /\  x A y ) )
86, 7bitr4i 244 . . . 4  |-  ( E* y  x A y  <->  E* y  e.  ran  A  x A y )
98ralbii 2721 . . 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 676 . 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 241 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 177    /\ wa 359    e. wcel 1725   E*wmo 2281   A.wral 2697   E*wrmo 2700   class class class wbr 4204   dom cdm 4870   ran crn 4871   Rel wrel 4875   Fun wfun 5440
This theorem is referenced by:  brdom4  8400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rmo 2705  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448
  Copyright terms: Public domain W3C validator