Theorem erexb 8512

Description: An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
erexb	⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))

Proof of Theorem erexb

Step	Hyp	Ref	Expression
1		dmexg 7742	. . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
2		erdm 8497	. . . 4 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
3	2	eleq1d 2823	. . 3 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V))
4	1, 3	syl5ib 243	. 2 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V))
5		erex 8511	. 2 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V))
6	4, 5	impbid 211	1 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  Vcvv 3431  dom cdm 5586   Er wer 8484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5076  df-opab 5138  df-xp 5592  df-rel 5593  df-cnv 5594  df-dm 5596  df-rn 5597  df-er 8487

This theorem is referenced by:  prtex  36881