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Theorem imadmrn 4702
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 2605 . . . . . . 7  |-  x  e. 
_V
2 vex 2605 . . . . . . 7  |-  y  e. 
_V
31, 2opeldm 4560 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
43pm4.71i 383 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  A  /\  x  e.  dom  A ) )
5 ancom 262 . . . . 5  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  dom  A )  <->  ( x  e.  dom  A  /\  <. x ,  y >.  e.  A
) )
64, 5bitr2i 183 . . . 4  |-  ( ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A )  <->  <. x ,  y >.  e.  A
)
76exbii 1537 . . 3  |-  ( E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
)  <->  E. x <. x ,  y >.  e.  A
)
87abbii 2195 . 2  |-  { y  |  E. x ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A ) }  =  { y  |  E. x <. x ,  y >.  e.  A }
9 dfima3 4695 . 2  |-  ( A
" dom  A )  =  { y  |  E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
) }
10 dfrn3 4546 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
118, 9, 103eqtr4i 2112 1  |-  ( A
" dom  A )  =  ran  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   <.cop 3403   dom cdm 4365   ran crn 4366   "cima 4368
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378
This theorem is referenced by:  cnvimarndm  4713  foima  5136  f1imacnv  5168  fsn2  5363  resfunexg  5408  funiunfvdm  5428  fnexALT  5765  uniqs2  6225  phplem4  6380  phplem4on  6392
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