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Theorem imadmrn 4891
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 2689 . . . . . . 7  |-  x  e. 
_V
2 vex 2689 . . . . . . 7  |-  y  e. 
_V
31, 2opeldm 4742 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
43pm4.71i 388 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  A  /\  x  e.  dom  A ) )
5 ancom 264 . . . . 5  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  dom  A )  <->  ( x  e.  dom  A  /\  <. x ,  y >.  e.  A
) )
64, 5bitr2i 184 . . . 4  |-  ( ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A )  <->  <. x ,  y >.  e.  A
)
76exbii 1584 . . 3  |-  ( E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
)  <->  E. x <. x ,  y >.  e.  A
)
87abbii 2255 . 2  |-  { y  |  E. x ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A ) }  =  { y  |  E. x <. x ,  y >.  e.  A }
9 dfima3 4884 . 2  |-  ( A
" dom  A )  =  { y  |  E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
) }
10 dfrn3 4728 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
118, 9, 103eqtr4i 2170 1  |-  ( A
" dom  A )  =  ran  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2125   <.cop 3530   dom cdm 4539   ran crn 4540   "cima 4542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552
This theorem is referenced by:  cnvimarndm  4903  foima  5350  f1imacnv  5384  fsn2  5594  resfunexg  5641  funiunfvdm  5664  fnexALT  6011  uniqs2  6489  mapsn  6584  phplem4  6749  phplem4on  6761  retopbas  12706
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