Theorem dmeqi 4898

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 4897	. 2 |- ( A = B -> dom A = dom B )
3	1, 2	ax-mp 5	1 |- dom A = dom B

Syntax hints:   = wceq 1373   dom cdm 4693

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-dm 4703

This theorem is referenced by:  dmxpm  4917  dmxpid  4918  dmxpin  4919  rncoss  4968  rncoeq  4971  rnun  5110  rnin  5111  rnxpm  5131  rnxpss  5133  imainrect  5147  dmpropg  5174  dmtpop  5177  rnsnopg  5180  fntpg  5349  fnreseql  5713  dmoprab  6049  reldmmpo  6080  elmpocl  6164  tfrlem8  6427  tfr2a  6430  tfrlemi14d  6442  tfr1onlemres  6458  tfri1dALT  6460  tfrcllemres  6471  xpassen  6950  sbthlemi5  7089  casedm  7214  djudm  7233  ctssdccl  7239  dmaddpi  7473  dmmulpi  7474  dmaddpq  7527  dmmulpq  7528  axaddf  8016  axmulf  8017  ennnfonelemom  12894  ennnfonelemdm  12906  structiedg0val  15754  isuhgrm  15782  isushgrm  15783  isupgren  15806  isumgren  15816