Theorem dmeqi 4830

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

Step	Hyp	Ref	Expression
1		dmeqi.1	. 2 ⊢ 𝐴 = 𝐵
2		dmeq 4829	. 2 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	ax-mp 5	1 ⊢ dom 𝐴 = dom 𝐵

Syntax hints:   = wceq 1353  dom cdm 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-dm 4638

This theorem is referenced by:  dmxpm  4849  dmxpid  4850  dmxpin  4851  rncoss  4899  rncoeq  4902  rnun  5039  rnin  5040  rnxpm  5060  rnxpss  5062  imainrect  5076  dmpropg  5103  dmtpop  5106  rnsnopg  5109  fntpg  5274  fnreseql  5628  dmoprab  5958  reldmmpo  5988  elmpocl  6071  tfrlem8  6321  tfr2a  6324  tfrlemi14d  6336  tfr1onlemres  6352  tfri1dALT  6354  tfrcllemres  6365  xpassen  6832  sbthlemi5  6962  casedm  7087  djudm  7106  ctssdccl  7112  dmaddpi  7326  dmmulpi  7327  dmaddpq  7380  dmmulpq  7381  axaddf  7869  axmulf  7870  ennnfonelemom  12411  ennnfonelemdm  12423