|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > erexb | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| erexb | ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dmexg 7923 | . . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V) | |
| 2 | erdm 8755 | . . . 4 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
| 3 | 2 | eleq1d 2826 | . . 3 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V)) | 
| 4 | 1, 3 | imbitrid 244 | . 2 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V)) | 
| 5 | erex 8769 | . 2 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V)) | |
| 6 | 4, 5 | impbid 212 | 1 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 Vcvv 3480 dom cdm 5685 Er wer 8742 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-er 8745 | 
| This theorem is referenced by: prtex 38881 | 
| Copyright terms: Public domain | W3C validator |