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Theorem funcnv3 5062
Description: A condition showing a class is single-rooted. (See funcnv 5061). (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 4612 . . . . . 6  |-  ran  A  =  { y  |  E. x  x A y }
21abeq2i 2198 . . . . 5  |-  ( y  e.  ran  A  <->  E. x  x A y )
32biimpi 118 . . . 4  |-  ( y  e.  ran  A  ->  E. x  x A
y )
43biantrurd 299 . . 3  |-  ( y  e.  ran  A  -> 
( E* x  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) ) )
54ralbiia 2392 . 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 5061 . 2  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
7 df-reu 2366 . . . 4  |-  ( E! x  e.  dom  A  x A y  <->  E! x
( x  e.  dom  A  /\  x A y ) )
8 vex 2622 . . . . . . 7  |-  x  e. 
_V
9 vex 2622 . . . . . . 7  |-  y  e. 
_V
108, 9breldm 4628 . . . . . 6  |-  ( x A y  ->  x  e.  dom  A )
1110pm4.71ri 384 . . . . 5  |-  ( x A y  <->  ( x  e.  dom  A  /\  x A y ) )
1211eubii 1957 . . . 4  |-  ( E! x  x A y  <-> 
E! x ( x  e.  dom  A  /\  x A y ) )
13 eu5 1995 . . . 4  |-  ( E! x  x A y  <-> 
( E. x  x A y  /\  E* x  x A y ) )
147, 12, 133bitr2i 206 . . 3  |-  ( E! x  e.  dom  A  x A y  <->  ( E. x  x A y  /\  E* x  x A
y ) )
1514ralbii 2384 . 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 210 1  |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1426    e. wcel 1438   E!weu 1948   E*wmo 1949   A.wral 2359   E!wreu 2361   class class class wbr 3837   `'ccnv 4427   dom cdm 4428   ran crn 4429   Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004
This theorem is referenced by: (None)
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