Theorem erth 3219

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 3215	. . . 4 |- (x = A -> [x]R = [A]R)
2	1	cleq1d 1109	. . 3 |- (x = A -> ([x]R = [B]R <-> [A]R = [B]R))
3		breq1 2065	. . 3 |- (x = A -> (xRB <-> ARB))
4	2, 3	bibi12d 477	. 2 |- (x = A -> (([x]R = [B]R <-> xRB) <-> ([A]R = [B]R <-> ARB)))
5		erth.2	. . . . 5 |- Er R
6	5	erref 3212	. . . 4 |- (x e. (dom R u. ran R) -> xRx)
7		eleq2 1150	. . . . . . 7 |- ([x]R = [B]R -> (x e. [x]R <-> x e. [B]R))
8		visset 1350	. . . . . . . 8 |- x e. V
9	8, 8	elec 3216	. . . . . . 7 |- (x e. [x]R <-> xRx)
10		erth.1	. . . . . . . 8 |- B e. V
11	8, 10	elec 3216	. . . . . . 7 |- (x e. [B]R <-> BRx)
12	7, 9, 11	3bitr3g 427	. . . . . 6 |- ([x]R = [B]R -> (xRx <-> BRx))
13	10, 8, 5	ersym 3209	. . . . . 6 |- (BRx -> xRB)
14	12, 13	syl6bi 187	. . . . 5 |- ([x]R = [B]R -> (xRx -> xRB))
15	14	com12 13	. . . 4 |- (xRx -> ([x]R = [B]R -> xRB))
16	6, 15	syl 12	. . 3 |- (x e. (dom R u. ran R) -> ([x]R = [B]R -> xRB))
17	8, 10, 5	erthi 3218	. . . 4 |- (xRB -> [x]R = [B]R)
18	17	a1i 7	. . 3 |- (x e. (dom R u. ran R) -> (xRB -> [x]R = [B]R))
19	16, 18	impbid 397	. 2 |- (x e. (dom R u. ran R) -> ([x]R = [B]R <-> xRB))
20	4, 19	vtoclga 1387	1 |- (A e. (dom R u. ran R) -> ([A]R = [B]R <-> ARB))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092  Vcvv 1348   u. cun 1485   class class class wbr 2054  dom cdm 2410  ran crn 2411  Er wer 3197  [cec 3198

This theorem is referenced by:  erthdm 3220

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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