MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imaeq1 Structured version   Unicode version

Theorem imaeq1 5330
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 5265 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 5228 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 5012 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 5012 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2533 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   ran crn 5000    |` cres 5001   "cima 5002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  imaeq1i  5332  imaeq1d  5334  suppval  6900  eceq2  7346  marypha1lem  7889  marypha1  7890  ackbij2lem2  8616  ackbij2lem3  8617  r1om  8620  limsupval  13256  isacs1i  14908  mreacs  14909  islindf  18614  iscnp  19504  xkoccn  19855  xkohaus  19889  xkoco1cn  19893  xkoco2cn  19894  xkococnlem  19895  xkococn  19896  xkoinjcn  19923  fmval  20179  fmf  20181  utoptop  20472  restutop  20475  restutopopn  20476  ustuqtoplem  20477  ustuqtop1  20479  ustuqtop2  20480  ustuqtop4  20482  ustuqtop5  20483  utopsnneiplem  20485  utopsnnei  20487  neipcfilu  20534  metutopOLD  20820  psmetutop  20821  cfilfval  21438  elply2  22328  coeeu  22357  coelem  22358  coeeq  22359  dmarea  23015  tailfval  29793  bj-cleq  33600
  Copyright terms: Public domain W3C validator