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Theorem funcnv3 5471
Description: A condition showing a class is single-rooted. (See funcnv 5470). (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 5018 . . . . . 6  |-  ran  A  =  { y  |  E. x  x A y }
21abeq2i 2511 . . . . 5  |-  ( y  e.  ran  A  <->  E. x  x A y )
32biimpi 187 . . . 4  |-  ( y  e.  ran  A  ->  E. x  x A
y )
43biantrurd 495 . . 3  |-  ( y  e.  ran  A  -> 
( E* x  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) ) )
54ralbiia 2698 . 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 5470 . 2  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
7 df-reu 2673 . . . 4  |-  ( E! x  e.  dom  A  x A y  <->  E! x
( x  e.  dom  A  /\  x A y ) )
8 vex 2919 . . . . . . 7  |-  x  e. 
_V
9 vex 2919 . . . . . . 7  |-  y  e. 
_V
108, 9breldm 5033 . . . . . 6  |-  ( x A y  ->  x  e.  dom  A )
1110pm4.71ri 615 . . . . 5  |-  ( x A y  <->  ( x  e.  dom  A  /\  x A y ) )
1211eubii 2263 . . . 4  |-  ( E! x  x A y  <-> 
E! x ( x  e.  dom  A  /\  x A y ) )
13 eu5 2292 . . . 4  |-  ( E! x  x A y  <-> 
( E. x  x A y  /\  E* x  x A y ) )
147, 12, 133bitr2i 265 . . 3  |-  ( E! x  e.  dom  A  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) )
1514ralbii 2690 . 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 269 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 177    /\ wa 359   E.wex 1547    e. wcel 1721   E!weu 2254   E*wmo 2255   A.wral 2666   E!wreu 2668   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   ran crn 4838   Fun wfun 5407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415
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