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Theorem imaeq1 5023
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 4965 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 4922 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4718 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4718 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2353 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   ran crn 4706    |` cres 4707   "cima 4708
This theorem is referenced by:  imaeq1i  5025  imaeq1d  5027  eceq2  6713  marypha1lem  7202  marypha1  7203  ackbij2lem2  7882  ackbij2lem3  7883  r1om  7886  limsupval  11964  isacs1i  13575  mreacs  13576  iscnp  16983  xkoccn  17329  xkohaus  17363  xkoco1cn  17367  xkoco2cn  17368  xkococnlem  17369  xkococn  17370  xkoinjcn  17397  fmval  17654  fmf  17656  cfilfval  18706  elply2  19594  coeeu  19623  coelem  19624  coeeq  19625  dmarea  20268  isibcg  26294  tailfval  26424  islindf  27385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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