Theorem dmeqi 4864

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 4863	. 2 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
3	1, 2	ax-mp 5	1 ⊢ dom 𝐴 = dom 𝐵

Syntax hints:   = wceq 1364  dom cdm 4660

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-dm 4670

This theorem is referenced by:  dmxpm  4883  dmxpid  4884  dmxpin  4885  rncoss  4933  rncoeq  4936  rnun  5075  rnin  5076  rnxpm  5096  rnxpss  5098  imainrect  5112  dmpropg  5139  dmtpop  5142  rnsnopg  5145  fntpg  5311  fnreseql  5669  dmoprab  6000  reldmmpo  6031  elmpocl  6115  tfrlem8  6373  tfr2a  6376  tfrlemi14d  6388  tfr1onlemres  6404  tfri1dALT  6406  tfrcllemres  6417  xpassen  6886  sbthlemi5  7022  casedm  7147  djudm  7166  ctssdccl  7172  dmaddpi  7387  dmmulpi  7388  dmaddpq  7441  dmmulpq  7442  axaddf  7930  axmulf  7931  ennnfonelemom  12568  ennnfonelemdm  12580