Theorem erthdm 5341

Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership in the domain instead of just the field.

Hypotheses

Ref	Expression
erthdm.1	|- B e. _V
erthdm.2	|- Er R

Assertion

Ref	Expression
erthdm	|- (A e. dom R -> ([A]R = [B]R <-> ARB))

Proof of Theorem erthdm

Step	Hyp	Ref	Expression
1		elun1 2771	. 2 |- (A e. dom R -> A e. (dom R u. ran R))
2		erthdm.1	. . 3 |- B e. _V
3		erthdm.2	. . 3 |- Er R
4	2, 3	erth 5340	. 2 |- (A e. (dom R u. ran R) -> ([A]R = [B]R <-> ARB))
5	1, 4	syl 12	1 |- (A e. dom R -> ([A]R = [B]R <-> ARB))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591   class class class wbr 3338  dom cdm 3986  ran crn 3987  Er wer 5315  [cec 5316

This theorem is referenced by:  erthdmr 5342  ereldm 5343  eceqopreq 5372  th3qlem1 5373  enqeceq 6199  enreceq 6329  erthdmg 15730

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-er 5318  df-ec 5320

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