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Theorem imaeq1 5007
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )

Proof of Theorem imaeq1
StepHypRef Expression
1 reseq1 4949 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 4906 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4702 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4702 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2340 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  imaeq1i  5009  imaeq1d  5011  eceq2  6697  marypha1lem  7186  marypha1  7187  ackbij2lem2  7866  ackbij2lem3  7867  r1om  7870  limsupval  11948  isacs1i  13559  mreacs  13560  iscnp  16967  xkoccn  17313  xkohaus  17347  xkoco1cn  17351  xkoco2cn  17352  xkococnlem  17353  xkococn  17354  xkoinjcn  17381  fmval  17638  fmf  17640  cfilfval  18690  elply2  19578  coeeu  19607  coelem  19608  coeeq  19609  dmarea  20252  isibcg  26191  tailfval  26321  islindf  27282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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