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Theorem imaeq1 5198
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 5140 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 5097 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4891 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4891 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2493 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   ran crn 4879    |` cres 4880   "cima 4881
This theorem is referenced by:  imaeq1i  5200  imaeq1d  5202  eceq2  6942  marypha1lem  7438  marypha1  7439  ackbij2lem2  8120  ackbij2lem3  8121  r1om  8124  limsupval  12268  isacs1i  13882  mreacs  13883  iscnp  17301  xkoccn  17651  xkohaus  17685  xkoco1cn  17689  xkoco2cn  17690  xkococnlem  17691  xkococn  17692  xkoinjcn  17719  fmval  17975  fmf  17977  utoptop  18264  restutop  18267  restutopopn  18268  ustuqtoplem  18269  ustuqtop1  18271  ustuqtop2  18272  ustuqtop4  18274  ustuqtop5  18275  utopsnneiplem  18277  utopsnnei  18279  neipcfilu  18326  metutopOLD  18612  psmetutop  18613  cfilfval  19217  elply2  20115  coeeu  20144  coelem  20145  coeeq  20146  dmarea  20796  tailfval  26401  islindf  27259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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