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 7097 | . . 3 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V) | |
2 | erdm 7752 | . . . 4 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
3 | 2 | eleq1d 2686 | . . 3 ⊢ (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V)) |
4 | 1, 3 | syl5ib 234 | . 2 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V)) |
5 | erex 7766 | . 2 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V)) | |
6 | 4, 5 | impbid 202 | 1 ⊢ (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 Vcvv 3200 dom cdm 5114 Er wer 7739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-er 7742 |
This theorem is referenced by: prtex 34165 |
Copyright terms: Public domain | W3C validator |