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  5629  dmoprab  5959  reldmmpo  5989  elmpocl  6072  tfrlem8  6322  tfr2a  6325  tfrlemi14d  6337  tfr1onlemres  6353  tfri1dALT  6355  tfrcllemres  6366  xpassen  6833  sbthlemi5  6963  casedm  7088  djudm  7107  ctssdccl  7113  dmaddpi  7327  dmmulpi  7328  dmaddpq  7381  dmmulpq  7382  axaddf  7870  axmulf  7871  ennnfonelemom  12412  ennnfonelemdm  12424