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Theorem imaeq1 5151
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 5087 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 5050 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4835 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4835 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2468 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   ran crn 4823    |` cres 4824   "cima 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835
This theorem is referenced by:  imaeq1i  5153  imaeq1d  5155  suppval  6903  eceq2  7385  marypha1lem  7926  marypha1  7927  ackbij2lem2  8651  ackbij2lem3  8652  r1om  8655  limsupval  13444  isacs1i  15269  mreacs  15270  islindf  19137  iscnp  20029  xkoccn  20410  xkohaus  20444  xkoco1cn  20448  xkoco2cn  20449  xkococnlem  20450  xkococn  20451  xkoinjcn  20478  fmval  20734  fmf  20736  utoptop  21027  restutop  21030  restutopopn  21031  ustuqtoplem  21032  ustuqtop1  21034  ustuqtop2  21035  ustuqtop4  21037  ustuqtop5  21038  utopsnneiplem  21040  utopsnnei  21042  neipcfilu  21089  metutopOLD  21375  psmetutop  21376  cfilfval  21993  elply2  22883  coeeu  22912  coelem  22913  coeeq  22914  dmarea  23611  mclsax  29768  tailfval  30587  bj-cleq  31071  brtrclfv2  35686
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