Theorem ereldm 5343

Description: Equality of equivalence classes implies equivalence of domain membership.

Hypotheses

Ref	Expression
ereldm.1	|- A e. _V
ereldm.2	|- B e. _V
ereldm.3	|- Er R
ereldm.4	|- dom R = D

Assertion

Ref	Expression
ereldm	|- ([A]R = [B]R -> (A e. D <-> B e. D))

Proof of Theorem ereldm

Step	Hyp	Ref	Expression
1		ereldm.2	. . . . . 6 |- B e. _V
2		ereldm.3	. . . . . 6 |- Er R
3	1, 2	erthdm 5341	. . . . 5 |- (A e. dom R -> ([A]R = [B]R <-> ARB))
4	3	biimpcd 172	. . . 4 |- ([A]R = [B]R -> (A e. dom R -> ARB))
5		ereldm.1	. . . . . 6 |- A e. _V
6	5, 1, 2	ersymb 5331	. . . . 5 |- (ARB <-> BRA)
7	1	breldm 4161	. . . . 5 |- (BRA -> B e. dom R)
8	6, 7	sylbi 216	. . . 4 |- (ARB -> B e. dom R)
9	4, 8	syl6 25	. . 3 |- ([A]R = [B]R -> (A e. dom R -> B e. dom R))
10	5, 1, 2	erthdmr 5342	. . . . 5 |- (B e. dom R -> ([A]R = [B]R <-> ARB))
11	10	biimpcd 172	. . . 4 |- ([A]R = [B]R -> (B e. dom R -> ARB))
12	5	breldm 4161	. . . 4 |- (ARB -> A e. dom R)
13	11, 12	syl6 25	. . 3 |- ([A]R = [B]R -> (B e. dom R -> A e. dom R))
14	9, 13	impbid 574	. 2 |- ([A]R = [B]R -> (A e. dom R <-> B e. dom R))
15		ereldm.4	. . 3 |- dom R = D
16	15	eleq2i 1961	. 2 |- (A e. dom R <-> A e. D)
17	15	eleq2i 1961	. 2 |- (B e. dom R <-> B e. D)
18	14, 16, 17	3bitr3g 613	1 |- ([A]R = [B]R -> (A e. D <-> B e. D))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  dom cdm 3986  Er wer 5315  [cec 5316

This theorem is referenced by:  brecop 5365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-er 5318  df-ec 5320

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