Theorem dmeqi 4700

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 4699	. 2 |- ( A = B -> dom A = dom B )
3	1, 2	ax-mp 7	1 |- dom A = dom B

Syntax hints:   = wceq 1314   dom cdm 4499

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-dm 4509

This theorem is referenced by:  dmxpm  4719  dmxpid  4720  dmxpin  4721  rncoss  4767  rncoeq  4770  rnun  4905  rnin  4906  rnxpm  4926  rnxpss  4928  imainrect  4942  dmpropg  4969  dmtpop  4972  rnsnopg  4975  fntpg  5137  fnreseql  5484  dmoprab  5806  reldmmpo  5836  elmpocl  5922  tfrlem8  6169  tfr2a  6172  tfrlemi14d  6184  tfr1onlemres  6200  tfri1dALT  6202  tfrcllemres  6213  xpassen  6677  sbthlemi5  6801  casedm  6923  djudm  6942  ctssdccl  6948  dmaddpi  7081  dmmulpi  7082  dmaddpq  7135  dmmulpq  7136  axaddf  7603  axmulf  7604  ennnfonelemom  11766  ennnfonelemdm  11778