Theorem dmeqi 4786

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 4785	. 2 |- ( A = B -> dom A = dom B )
3	1, 2	ax-mp 5	1 |- dom A = dom B

Syntax hints:   = wceq 1335   dom cdm 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-dm 4595

This theorem is referenced by:  dmxpm  4805  dmxpid  4806  dmxpin  4807  rncoss  4855  rncoeq  4858  rnun  4993  rnin  4994  rnxpm  5014  rnxpss  5016  imainrect  5030  dmpropg  5057  dmtpop  5060  rnsnopg  5063  fntpg  5225  fnreseql  5576  dmoprab  5899  reldmmpo  5929  elmpocl  6015  tfrlem8  6262  tfr2a  6265  tfrlemi14d  6277  tfr1onlemres  6293  tfri1dALT  6295  tfrcllemres  6306  xpassen  6772  sbthlemi5  6902  casedm  7025  djudm  7044  ctssdccl  7050  dmaddpi  7240  dmmulpi  7241  dmaddpq  7294  dmmulpq  7295  axaddf  7783  axmulf  7784  ennnfonelemom  12124  ennnfonelemdm  12136