Theorem ereldm 3222

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

Hypotheses

Ref	Expression
ereldm.1	⊢ A ∈ V
ereldm.2	⊢ B ∈ V
ereldm.3	⊢ Er R
ereldm.4	⊢ dom R = D

Assertion

Ref	Expression
ereldm	⊢ ([A]R = [B]R → (A ∈ D ↔ B ∈ D))

Proof of Theorem ereldm

Step	Hyp	Ref	Expression
1		ereldm.2	. . . . . 6 ⊢ B ∈ V
2		ereldm.3	. . . . . 6 ⊢ Er R
3	1, 2	erthdm 3220	. . . . 5 ⊢ (A ∈ dom R → ([A]R = [B]R ↔ ARB))
4	3	biimpcd 137	. . . 4 ⊢ ([A]R = [B]R → (A ∈ dom R → ARB))
5		ereldm.1	. . . . . 6 ⊢ A ∈ V
6	5, 1, 2	ersymb 3210	. . . . 5 ⊢ (ARB ↔ BRA)
7	1	breldm 2535	. . . . 5 ⊢ (BRA → B ∈ dom R)
8	6, 7	sylbi 174	. . . 4 ⊢ (ARB → B ∈ dom R)
9	4, 8	syl6 23	. . 3 ⊢ ([A]R = [B]R → (A ∈ dom R → B ∈ dom R))
10	5, 1, 2	erthdmr 3221	. . . . 5 ⊢ (B ∈ dom R → ([A]R = [B]R ↔ ARB))
11	10	biimpcd 137	. . . 4 ⊢ ([A]R = [B]R → (B ∈ dom R → ARB))
12	5	breldm 2535	. . . 4 ⊢ (ARB → A ∈ dom R)
13	11, 12	syl6 23	. . 3 ⊢ ([A]R = [B]R → (B ∈ dom R → A ∈ dom R))
14	9, 13	impbid 397	. 2 ⊢ ([A]R = [B]R → (A ∈ dom R ↔ B ∈ dom R))
15		ereldm.4	. . 3 ⊢ dom R = D
16	15	eleq2i 1153	. 2 ⊢ (A ∈ dom R ↔ A ∈ D)
17	15	eleq2i 1153	. 2 ⊢ (B ∈ dom R ↔ B ∈ D)
18	14, 16, 17	3bitr3g 427	1 ⊢ ([A]R = [B]R → (A ∈ D ↔ B ∈ D))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  dom cdm 2410  Er wer 3197  [cec 3198

This theorem is referenced by:  brecop 3242

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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