Theorem dmeqi 4867

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 4866	. 2 |- ( A = B -> dom A = dom B )
3	1, 2	ax-mp 5	1 |- dom A = dom B

Syntax hints:   = wceq 1364   dom cdm 4663

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-dm 4673

This theorem is referenced by:  dmxpm  4886  dmxpid  4887  dmxpin  4888  rncoss  4936  rncoeq  4939  rnun  5078  rnin  5079  rnxpm  5099  rnxpss  5101  imainrect  5115  dmpropg  5142  dmtpop  5145  rnsnopg  5148  fntpg  5314  fnreseql  5672  dmoprab  6003  reldmmpo  6034  elmpocl  6118  tfrlem8  6376  tfr2a  6379  tfrlemi14d  6391  tfr1onlemres  6407  tfri1dALT  6409  tfrcllemres  6420  xpassen  6889  sbthlemi5  7027  casedm  7152  djudm  7171  ctssdccl  7177  dmaddpi  7392  dmmulpi  7393  dmaddpq  7446  dmmulpq  7447  axaddf  7935  axmulf  7936  ennnfonelemom  12625  ennnfonelemdm  12637