Theorem erthdm 3220

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 ∈ V
erthdm.2	⊢ Er R

Assertion

Ref	Expression
erthdm	⊢ (A ∈ dom R → ([A]R = [B]R ↔ ARB))

Proof of Theorem erthdm

Step	Hyp	Ref	Expression
1		elun1 1625	. 2 ⊢ (A ∈ dom R → A ∈ (dom R ∪ ran R))
2		erthdm.1	. . 3 ⊢ B ∈ V
3		erthdm.2	. . 3 ⊢ Er R
4	2, 3	erth 3219	. 2 ⊢ (A ∈ (dom R ∪ ran R) → ([A]R = [B]R ↔ ARB))
5	1, 4	syl 12	1 ⊢ (A ∈ dom R → ([A]R = [B]R ↔ ARB))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   class class class wbr 2054  dom cdm 2410  ran crn 2411  Er wer 3197  [cec 3198

This theorem is referenced by:  erthdmr 3221  ereldm 3222  eceqopreq 3249  th3qlem1 3250  enqeceq 3841  enreceq 3971

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-er 3200  df-ec 3202

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