Theorem erexb 8724

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 7890	. . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
2		erdm 8709	. . . 4 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
3	2	eleq1d 2818	. . 3 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V))
4	1, 3	imbitrid 243	. 2 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V))
5		erex 8723	. 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 3474  dom cdm 5675   Er wer 8696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-er 8699

This theorem is referenced by:  prtex  37738