Theorem erthdm 4280

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 2195	. 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 4279	. 2 |- (A e. (dom R u. ran R) -> ([A]R = [B]R <-> ARB))
5	1, 4	syl 10	1 |- (A e. dom R -> ([A]R = [B]R <-> ARB))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  Vcvv 1809   u. cun 2043   class class class wbr 2616  dom cdm 3167  ran crn 3168  Er wer 4255  [cec 4256

This theorem is referenced by:  erthdmr 4281  ereldm 4282  eceqopreq 4310  th3qlem1 4311  enqeceq 5034  enreceq 5164

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-xp 3181  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-er 4258  df-ec 4260

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