![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dmi | Structured version Visualization version GIF version |
Description: The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmi | ⊢ dom I = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqv 3478 | . 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I ) | |
2 | ax6ev 1966 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑥 | |
3 | vex 3473 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5849 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | equcom 2014 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
6 | 4, 5 | bitri 275 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥) |
7 | 6 | exbii 1843 | . . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥) |
8 | 2, 7 | mpbir 230 | . . 3 ⊢ ∃𝑦 𝑥 I 𝑦 |
9 | vex 3473 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | 9 | eldm 5897 | . . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦) |
11 | 8, 10 | mpbir 230 | . 2 ⊢ 𝑥 ∈ dom I |
12 | 1, 11 | mpgbir 1794 | 1 ⊢ dom I = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3469 class class class wbr 5142 I cid 5569 dom cdm 5672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-dm 5682 |
This theorem is referenced by: dmv 5919 dmresi 6049 idfn 6677 iprc 7913 |
Copyright terms: Public domain | W3C validator |