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Theorem imadmrn 5288
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 3061 . . . . . . 7  |-  x  e. 
_V
2 vex 3061 . . . . . . 7  |-  y  e. 
_V
31, 2opeldm 5148 . . . . . 6  |-  ( <.
x ,  y >.  e.  A  ->  x  e. 
dom  A )
43pm4.71i 630 . . . . 5  |-  ( <.
x ,  y >.  e.  A  <->  ( <. x ,  y >.  e.  A  /\  x  e.  dom  A ) )
5 ancom 448 . . . . 5  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  dom  A )  <->  ( x  e.  dom  A  /\  <. x ,  y >.  e.  A
) )
64, 5bitr2i 250 . . . 4  |-  ( ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A )  <->  <. x ,  y >.  e.  A
)
76exbii 1688 . . 3  |-  ( E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
)  <->  E. x <. x ,  y >.  e.  A
)
87abbii 2536 . 2  |-  { y  |  E. x ( x  e.  dom  A  /\  <. x ,  y
>.  e.  A ) }  =  { y  |  E. x <. x ,  y >.  e.  A }
9 dfima3 5281 . 2  |-  ( A
" dom  A )  =  { y  |  E. x ( x  e. 
dom  A  /\  <. x ,  y >.  e.  A
) }
10 dfrn3 5134 . 2  |-  ran  A  =  { y  |  E. x <. x ,  y
>.  e.  A }
118, 9, 103eqtr4i 2441 1  |-  ( A
" dom  A )  =  ran  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   {cab 2387   <.cop 3977   dom cdm 4942   ran crn 4943   "cima 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4948  df-cnv 4950  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955
This theorem is referenced by:  cnvimarndm  5299  foima  5739  f1imacnv  5771  fsn2  6005  resfunexg  6074  elunirnALT  6101  fnexALT  6704  uniqs2  7330  mapsn  7418  phplem4  7657  php3  7661  jech9.3  8184  fin4en1  8641  retopbas  21451  plyeq0  22792  rnelshi  27271
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