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Theorem dmeqi 4604
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 4603 . 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 1287  dom cdm 4410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-dm 4420
This theorem is referenced by:  dmxpm  4623  dmxpinm  4624  rncoss  4670  rncoeq  4673  rnun  4803  rnin  4804  rnxpm  4823  rnxpss  4825  imainrect  4839  dmpropg  4866  dmtpop  4869  rnsnopg  4872  fntpg  5032  fnreseql  5366  dmoprab  5680  reldmmpt2  5707  elmpt2cl  5793  tfrlem8  6031  tfr2a  6034  tfrlemi14d  6046  tfr1onlemres  6062  tfri1dALT  6064  tfrcllemres  6075  xpassen  6492  sbthlemi5  6607  casedm  6714  djudm  6722  dmaddpi  6821  dmmulpi  6822  dmaddpq  6875  dmmulpq  6876
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