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Theorem dffun9 4958
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 4956 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x  e.  dom  A E* y  x A y ) )
2 vex 2577 . . . . . . . 8  |-  x  e. 
_V
3 vex 2577 . . . . . . . 8  |-  y  e. 
_V
42, 3brelrn 4595 . . . . . . 7  |-  ( x A y  ->  y  e.  ran  A )
54pm4.71ri 378 . . . . . 6  |-  ( x A y  <->  ( y  e.  ran  A  /\  x A y ) )
65mobii 1953 . . . . 5  |-  ( E* y  x A y  <->  E* y ( y  e. 
ran  A  /\  x A y ) )
7 df-rmo 2331 . . . . 5  |-  ( E* y  e.  ran  A  x A y  <->  E* y
( y  e.  ran  A  /\  x A y ) )
86, 7bitr4i 180 . . . 4  |-  ( E* y  x A y  <->  E* y  e.  ran  A  x A y )
98ralbii 2347 . . 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 438 . 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 177 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 101    <-> wb 102    e. wcel 1409   E*wmo 1917   A.wral 2323   E*wrmo 2326   class class class wbr 3792   dom cdm 4373   ran crn 4374   Rel wrel 4378   Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rmo 2331  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932
This theorem is referenced by: (None)
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