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Theorem imadmrn 4976
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 2740 . . . . . . 7  |-  x  e. 
_V
2 vex 2740 . . . . . . 7  |-  y  e. 
_V
31, 2opeldm 4826 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
43pm4.71i 391 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  A  /\  x  e.  dom  A ) )
5 ancom 266 . . . . 5  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  dom  A )  <->  ( x  e.  dom  A  /\  <. x ,  y >.  e.  A
) )
64, 5bitr2i 185 . . . 4  |-  ( ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A )  <->  <. x ,  y >.  e.  A
)
76exbii 1605 . . 3  |-  ( E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
)  <->  E. x <. x ,  y >.  e.  A
)
87abbii 2293 . 2  |-  { y  |  E. x ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A ) }  =  { y  |  E. x <. x ,  y >.  e.  A }
9 dfima3 4969 . 2  |-  ( A
" dom  A )  =  { y  |  E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
) }
10 dfrn3 4812 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
118, 9, 103eqtr4i 2208 1  |-  ( A
" dom  A )  =  ran  A
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   <.cop 3594   dom cdm 4623   ran crn 4624   "cima 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636
This theorem is referenced by:  cnvimarndm  4988  foima  5439  f1imacnv  5474  fsn2  5686  resfunexg  5733  funiunfvdm  5758  fnexALT  6106  uniqs2  6589  mapsn  6684  phplem4  6849  phplem4on  6861  retopbas  13683
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