MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imadmrn Unicode version

Theorem imadmrn 5040
Description: The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imadmrn  |-  ( A
" dom  A )  =  ran  A

Proof of Theorem imadmrn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . . 7  |-  x  e. 
_V
2 vex 2804 . . . . . . 7  |-  y  e. 
_V
31, 2opeldm 4898 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
43pm4.71i 613 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  A  /\  x  e.  dom  A ) )
5 ancom 437 . . . . 5  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  dom  A )  <->  ( x  e.  dom  A  /\  <. x ,  y >.  e.  A
) )
64, 5bitr2i 241 . . . 4  |-  ( ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A )  <->  <. x ,  y >.  e.  A
)
76exbii 1572 . . 3  |-  ( E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
)  <->  E. x <. x ,  y >.  e.  A
)
87abbii 2408 . 2  |-  { y  |  E. x ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A ) }  =  { y  |  E. x <. x ,  y >.  e.  A }
9 dfima3 5031 . 2  |-  ( A
" dom  A )  =  { y  |  E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
) }
10 dfrn3 4885 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
118, 9, 103eqtr4i 2326 1  |-  ( A
" dom  A )  =  ran  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   <.cop 3656   dom cdm 4705   ran crn 4706   "cima 4708
This theorem is referenced by:  cnvimarndm  5050  foima  5472  f1imacnv  5505  fsn2  5714  resfunexg  5753  fnexALT  5758  elunirn  5793  uniqs2  6737  mapsn  6825  phplem4  7059  php3  7063  jech9.3  7502  fin4en1  7951  retopbas  18285  plyeq0  19609  rnelshi  22655  smbkle  26146  cndpv  26152  pgapspf  26155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
  Copyright terms: Public domain W3C validator