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Theorem funcnv3 5649
Description: A condition showing a class is single-rooted. (See funcnv 5648). (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 5191 . . . . . 6  |-  ran  A  =  { y  |  E. x  x A y }
21abeq2i 2594 . . . . 5  |-  ( y  e.  ran  A  <->  E. x  x A y )
32biimpi 194 . . . 4  |-  ( y  e.  ran  A  ->  E. x  x A
y )
43biantrurd 508 . . 3  |-  ( y  e.  ran  A  -> 
( E* x  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) ) )
54ralbiia 2894 . 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 5648 . 2  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
7 df-reu 2821 . . . 4  |-  ( E! x  e.  dom  A  x A y  <->  E! x
( x  e.  dom  A  /\  x A y ) )
8 vex 3116 . . . . . . 7  |-  x  e. 
_V
9 vex 3116 . . . . . . 7  |-  y  e. 
_V
108, 9breldm 5207 . . . . . 6  |-  ( x A y  ->  x  e.  dom  A )
1110pm4.71ri 633 . . . . 5  |-  ( x A y  <->  ( x  e.  dom  A  /\  x A y ) )
1211eubii 2300 . . . 4  |-  ( E! x  x A y  <-> 
E! x ( x  e.  dom  A  /\  x A y ) )
13 eu5 2305 . . . 4  |-  ( E! x  x A y  <-> 
( E. x  x A y  /\  E* x  x A y ) )
147, 12, 133bitr2i 273 . . 3  |-  ( E! x  e.  dom  A  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) )
1514ralbii 2895 . 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 277 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1596    e. wcel 1767   E!weu 2275   E*wmo 2276   A.wral 2814   E!wreu 2816   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   ran crn 5000   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5590
This theorem is referenced by: (None)
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