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Theorem imaeq1 5424
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem imaeq1
StepHypRef Expression
1 reseq1 5354 . . 3 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
21rneqd 5317 . 2 (𝐴 = 𝐵 → ran (𝐴𝐶) = ran (𝐵𝐶))
3 df-ima 5092 . 2 (𝐴𝐶) = ran (𝐴𝐶)
4 df-ima 5092 . 2 (𝐵𝐶) = ran (𝐵𝐶)
52, 3, 43eqtr4g 2685 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  ran crn 5080  cres 5081  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by:  imaeq1i  5426  imaeq1d  5428  suppval  7243  eceq2  7730  marypha1lem  8284  marypha1  8285  ackbij2lem2  9007  ackbij2lem3  9008  r1om  9011  limsupval  14134  isacs1i  16234  mreacs  16235  islindf  20065  iscnp  20946  xkoccn  21327  xkohaus  21361  xkoco1cn  21365  xkoco2cn  21366  xkococnlem  21367  xkococn  21368  xkoinjcn  21395  fmval  21652  fmf  21654  utoptop  21943  restutop  21946  restutopopn  21947  ustuqtoplem  21948  ustuqtop1  21950  ustuqtop2  21951  ustuqtop4  21953  ustuqtop5  21954  utopsnneiplem  21956  utopsnnei  21958  neipcfilu  22005  psmetutop  22277  cfilfval  22965  elply2  23851  coeeu  23880  coelem  23881  coeeq  23882  dmarea  24579  mclsax  31166  tailfval  32001  bj-cleq  32588  poimirlem15  33042  poimirlem24  33051  brtrclfv2  37486
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