Theorem ereldm 4285

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 4283	. . . . 5 |- (A e. dom R -> ([A]R = [B]R <-> ARB))
4	3	biimpcd 155	. . . 4 |- ([A]R = [B]R -> (A e. dom R -> ARB))
5		ereldm.1	. . . . . 6 |- A e. V
6	5, 1, 2	ersymb 4273	. . . . 5 |- (ARB <-> BRA)
7	1	breldm 3315	. . . . 5 |- (BRA -> B e. dom R)
8	6, 7	sylbi 199	. . . 4 |- (ARB -> B e. dom R)
9	4, 8	syl6 22	. . 3 |- ([A]R = [B]R -> (A e. dom R -> B e. dom R))
10	5, 1, 2	erthdmr 4284	. . . . 5 |- (B e. dom R -> ([A]R = [B]R <-> ARB))
11	10	biimpcd 155	. . . 4 |- ([A]R = [B]R -> (B e. dom R -> ARB))
12	5	breldm 3315	. . . 4 |- (ARB -> A e. dom R)
13	11, 12	syl6 22	. . 3 |- ([A]R = [B]R -> (B e. dom R -> A e. dom R))
14	9, 13	impbid 516	. 2 |- ([A]R = [B]R -> (A e. dom R <-> B e. dom R))
15		ereldm.4	. . 3 |- dom R = D
16	15	eleq2i 1538	. 2 |- (A e. dom R <-> A e. D)
17	15	eleq2i 1538	. 2 |- (B e. dom R <-> B e. D)
18	14, 16, 17	3bitr3g 554	1 |- ([A]R = [B]R -> (A e. D <-> B e. D))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  dom cdm 3170  Er wer 4258  [cec 4259

This theorem is referenced by:  brecop 4306

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-er 4261  df-ec 4263

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