Theorem erthi 3218

Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57.

Hypotheses

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

Assertion

Ref	Expression
erthi	|- (ARB -> [A]R = [B]R)

Proof of Theorem erthi

Step	Hyp	Ref	Expression
1		erthi.1	. . . . . 6 |- A e. V
2		erthi.2	. . . . . 6 |- B e. V
3		erthi.3	. . . . . 6 |- Er R
4	1, 2, 3	ersymb 3210	. . . . 5 |- (ARB <-> BRA)
5		visset 1350	. . . . . . 7 |- x e. V
6	2, 1, 5, 3	ertr 3211	. . . . . 6 |- ((BRA /\ ARx) -> BRx)
7	6	exp 291	. . . . 5 |- (BRA -> (ARx -> BRx))
8	4, 7	sylbi 174	. . . 4 |- (ARB -> (ARx -> BRx))
9	1, 2, 5, 3	ertr 3211	. . . . 5 |- ((ARB /\ BRx) -> ARx)
10	9	exp 291	. . . 4 |- (ARB -> (BRx -> ARx))
11	8, 10	impbid 397	. . 3 |- (ARB -> (ARx <-> BRx))
12	5, 1	elec 3216	. . 3 |- (x e. [A]R <-> ARx)
13	5, 2	elec 3216	. . 3 |- (x e. [B]R <-> BRx)
14	11, 12, 13	3bitr4g 428	. 2 |- (ARB -> (x e. [A]R <-> x e. [B]R))
15	14	cleqrd 1100	1 |- (ARB -> [A]R = [B]R)

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  Er wer 3197  [cec 3198

This theorem is referenced by:  erth 3219  erdisj 3223  th3qlem1 3250  distrpqlem 3860

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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