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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 7476 | . . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V) | |
2 | erdm 8156 | . . . 4 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
3 | 2 | eleq1d 2869 | . . 3 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V)) |
4 | 1, 3 | syl5ib 245 | . 2 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V)) |
5 | erex 8170 | . 2 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V)) | |
6 | 4, 5 | impbid 213 | 1 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2083 Vcvv 3440 dom cdm 5450 Er wer 8143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458 df-dm 5460 df-rn 5461 df-er 8146 |
This theorem is referenced by: prtex 35568 |
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