Theorem erthdmr 4290

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

Hypotheses

Ref	Expression
erthdmr.1	|- A e. V
erthdmr.2	|- B e. V
erthdmr.3	|- Er R

Assertion

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

Proof of Theorem erthdmr

Step	Hyp	Ref	Expression
1		erthdmr.1	. . 3 |- A e. V
2		erthdmr.3	. . 3 |- Er R
3	1, 2	erthdm 4289	. 2 |- (B e. dom R -> ([B]R = [A]R <-> BRA))
4		eqcom 1480	. 2 |- ([A]R = [B]R <-> [B]R = [A]R)
5		erthdmr.2	. . 3 |- B e. V
6	1, 5, 2	ersymb 4279	. 2 |- (ARB <-> BRA)
7	3, 4, 6	3bitr4g 557	1 |- (B e. dom R -> ([A]R = [B]R <-> ARB))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814   class class class wbr 2624  dom cdm 3176  Er wer 4264  [cec 4265

This theorem is referenced by:  ereldm 4291

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-er 4267  df-ec 4269

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