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Theorem funcnv3 5655
Description: A condition showing a class is single-rooted. (See funcnv 5654). (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 5201 . . . . . 6  |-  ran  A  =  { y  |  E. x  x A y }
21abeq2i 2584 . . . . 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 2887 . 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 5654 . 2  |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
7 df-reu 2814 . . . 4  |-  ( E! x  e.  dom  A  x A y  <->  E! x
( x  e.  dom  A  /\  x A y ) )
8 vex 3112 . . . . . . 7  |-  x  e. 
_V
9 vex 3112 . . . . . . 7  |-  y  e. 
_V
108, 9breldm 5217 . . . . . 6  |-  ( x A y  ->  x  e.  dom  A )
1110pm4.71ri 633 . . . . 5  |-  ( x A y  <->  ( x  e.  dom  A  /\  x A y ) )
1211eubii 2307 . . . 4  |-  ( E! x  x A y  <-> 
E! x ( x  e.  dom  A  /\  x A y ) )
13 eu5 2311 . . . 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 2888 . 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 1613    e. wcel 1819   E!weu 2283   E*wmo 2284   A.wral 2807   E!wreu 2809   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596
This theorem is referenced by: (None)
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