Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmeqi GIF version

Theorem dmeqi 4536
 Description: Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
Hypothesis
Ref Expression
dmeqi.1 𝐴 = 𝐵
Assertion
Ref Expression
dmeqi dom 𝐴 = dom 𝐵

Proof of Theorem dmeqi
StepHypRef Expression
1 dmeqi.1 . 2 𝐴 = 𝐵
2 dmeq 4535 . 2 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
31, 2ax-mp 7 1 dom 𝐴 = dom 𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1243  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355 This theorem is referenced by:  dmxpm  4555  dmxpinm  4556  rncoss  4602  rncoeq  4605  rnun  4732  rnin  4733  rnxpm  4752  rnxpss  4754  imainrect  4766  dmpropg  4793  dmtpop  4796  rnsnopg  4799  fntpg  4955  fnreseql  5277  dmoprab  5585  reldmmpt2  5612  elmpt2cl  5698  tfrlem8  5934  tfr2a  5936  tfrlemi14d  5947  xpassen  6304  dmaddpi  6423  dmmulpi  6424  dmaddpq  6477  dmmulpq  6478
 Copyright terms: Public domain W3C validator