Theorem erth 4340

Description: Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82.

Hypotheses

Ref	Expression
erth.1	|- B e. V
erth.2	|- Er R

Assertion

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

Proof of Theorem erth

Step	Hyp	Ref	Expression
1		eceq2 4336	. . . 4 |- (x = A -> [x]R = [A]R)
2	1	eqeq1d 1530	. . 3 |- (x = A -> ([x]R = [B]R <-> [A]R = [B]R))
3		breq1 2677	. . 3 |- (x = A -> (xRB <-> ARB))
4	2, 3	bibi12d 640	. 2 |- (x = A -> (([x]R = [B]R <-> xRB) <-> ([A]R = [B]R <-> ARB)))
5		eleq2 1582	. . . . . 6 |- ([x]R = [B]R -> (x e. [x]R <-> x e. [B]R))
6		visset 1860	. . . . . . 7 |- x e. V
7	6, 6	elec 4337	. . . . . 6 |- (x e. [x]R <-> xRx)
8		erth.1	. . . . . . 7 |- B e. V
9	6, 8	elec 4337	. . . . . 6 |- (x e. [B]R <-> BRx)
10	5, 7, 9	3bitr3g 565	. . . . 5 |- ([x]R = [B]R -> (xRx <-> BRx))
11		erth.2	. . . . . 6 |- Er R
12	8, 6, 11	ersym 4330	. . . . 5 |- (BRx -> xRB)
13	10, 12	syl6bi 221	. . . 4 |- ([x]R = [B]R -> (xRx -> xRB))
14	11	erref 4333	. . . 4 |- (x e. (dom R u. ran R) -> xRx)
15	13, 14	syl5com 52	. . 3 |- (x e. (dom R u. ran R) -> ([x]R = [B]R -> xRB))
16	6, 8, 11	erthi 4339	. . 3 |- (xRB -> [x]R = [B]R)
17	15, 16	impbid1 528	. 2 |- (x e. (dom R u. ran R) -> ([x]R = [B]R <-> xRB))
18	4, 17	vtoclga 1899	1 |- (A e. (dom R u. ran R) -> ([A]R = [B]R <-> ARB))

Syntax hints:   -> wi 3   <-> wb 153   = wceq 997   e. wcel 999  Vcvv 1858   u. cun 2096   class class class wbr 2674  dom cdm 3227  ran crn 3228  Er wer 4316  [cec 4317

This theorem is referenced by:  erthdm 4341

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-rex 1697  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-br 2675  df-opab 2722  df-xp 3241  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-er 4319  df-ec 4321

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