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Theorem funcnv3 5393
Description: A condition showing a class is single-rooted. (See funcnv 5392). (Contributed by NM, 26-May-2006.)
Assertion
Ref Expression
funcnv3  |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
Distinct variable group:    x, y, A

Proof of Theorem funcnv3
StepHypRef Expression
1 dfrn2 4950 . . . . . 6  |-  ran  A  =  { y  |  E. x  x A y }
21abeq2i 2465 . . . . 5  |-  ( y  e.  ran  A  <->  E. x  x A y )
32biimpi 186 . . . 4  |-  ( y  e.  ran  A  ->  E. x  x A
y )
43biantrurd 494 . . 3  |-  ( y  e.  ran  A  -> 
( E* x  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) ) )
54ralbiia 2651 . 2  |-  ( A. y  e.  ran  A E* x  x A y  <->  A. y  e.  ran  A ( E. x  x A y  /\  E* x  x A y ) )
6 funcnv 5392 . 2  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
7 df-reu 2626 . . . 4  |-  ( E! x  e.  dom  A  x A y  <->  E! x
( x  e.  dom  A  /\  x A y ) )
8 vex 2867 . . . . . . 7  |-  x  e. 
_V
9 vex 2867 . . . . . . 7  |-  y  e. 
_V
108, 9breldm 4965 . . . . . 6  |-  ( x A y  ->  x  e.  dom  A )
1110pm4.71ri 614 . . . . 5  |-  ( x A y  <->  ( x  e.  dom  A  /\  x A y ) )
1211eubii 2218 . . . 4  |-  ( E! x  x A y  <-> 
E! x ( x  e.  dom  A  /\  x A y ) )
13 eu5 2247 . . . 4  |-  ( E! x  x A y  <-> 
( E. x  x A y  /\  E* x  x A y ) )
147, 12, 133bitr2i 264 . . 3  |-  ( E! x  e.  dom  A  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) )
1514ralbii 2643 . 2  |-  ( A. y  e.  ran  A E! x  e.  dom  A  x A y  <->  A. y  e.  ran  A ( E. x  x A y  /\  E* x  x A y ) )
165, 6, 153bitr4i 268 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1541    e. wcel 1710   E!weu 2209   E*wmo 2210   A.wral 2619   E!wreu 2621   class class class wbr 4104   `'ccnv 4770   dom cdm 4771   ran crn 4772   Fun wfun 5331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-fun 5339
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