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Theorem imaeq1 5152
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 5091 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 5054 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4840 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4840 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2490 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362   ran crn 4828    |` cres 4829   "cima 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-cnv 4835  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840
This theorem is referenced by:  imaeq1i  5154  imaeq1d  5156  suppval  6681  eceq2  7126  marypha1lem  7671  marypha1  7672  ackbij2lem2  8397  ackbij2lem3  8398  r1om  8401  limsupval  12936  isacs1i  14578  mreacs  14579  islindf  18083  iscnp  18683  xkoccn  19034  xkohaus  19068  xkoco1cn  19072  xkoco2cn  19073  xkococnlem  19074  xkococn  19075  xkoinjcn  19102  fmval  19358  fmf  19360  utoptop  19651  restutop  19654  restutopopn  19655  ustuqtoplem  19656  ustuqtop1  19658  ustuqtop2  19659  ustuqtop4  19661  ustuqtop5  19662  utopsnneiplem  19664  utopsnnei  19666  neipcfilu  19713  metutopOLD  19999  psmetutop  20000  cfilfval  20617  elply2  21549  coeeu  21578  coelem  21579  coeeq  21580  dmarea  22236  tailfval  28437  bj-cleq  32058
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