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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 3470 | . 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I ) | |
2 | ax6ev 1965 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑥 | |
3 | vex 3465 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5855 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | equcom 2013 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
6 | 4, 5 | bitri 274 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥) |
7 | 6 | exbii 1842 | . . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥) |
8 | 2, 7 | mpbir 230 | . . 3 ⊢ ∃𝑦 𝑥 I 𝑦 |
9 | vex 3465 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | 9 | eldm 5903 | . . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦) |
11 | 8, 10 | mpbir 230 | . 2 ⊢ 𝑥 ∈ dom I |
12 | 1, 11 | mpgbir 1793 | 1 ⊢ dom I = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3461 class class class wbr 5149 I cid 5575 dom cdm 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-dm 5688 |
This theorem is referenced by: dmv 5925 dmresi 6056 idfn 6684 iprc 7919 |
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