MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imaeq1i Structured version   Visualization version   GIF version

Theorem imaeq1i 6056
Description: Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
Hypothesis
Ref Expression
imaeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
imaeq1i (𝐴𝐶) = (𝐵𝐶)

Proof of Theorem imaeq1i
StepHypRef Expression
1 imaeq1i.1 . 2 𝐴 = 𝐵
2 imaeq1 6054 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
31, 2ax-mp 5 1 (𝐴𝐶) = (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  mptpreima  6237  csbpredg  6306  isarep2  6639  suppun  8174  suppco  8197  fsuppun  9388  fsuppcolem  9402  marypha2lem4  9439  dfoi  9512  r1limg  9772  isf34lem3  10376  compss  10377  fpwwe2lem12  10643  infrenegsup  12204  gsumzf1o  19828  ssidcn  23079  cnco  23090  qtopres  23522  idqtop  23530  qtopcn  23538  mbfid  25484  mbfres  25493  cncombf  25507  dvlog  26499  efopnlem2  26505  eucrct2eupth  29932  disjpreima  32249  imadifxp  32266  rinvf1o  32288  cyc3genpm  32748  mbfmcst  33723  mbfmco  33728  sitmcl  33815  eulerpartlemt  33835  eulerpartlemmf  33839  eulerpart  33846  0rrv  33915  mclsppslem  35039  bj-iminvid  36542  mptsnun  36686  poimirlem3  36957  ftc1anclem3  37029  areacirclem5  37046  cytpval  42416  arearect  42429  brtrclfv2  42943  0cnf  45054  fourierdlem62  45345  smfco  45979
  Copyright terms: Public domain W3C validator