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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 3490 | . 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I ) | |
| 2 | ax6ev 1969 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑥 | |
| 3 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 4 | 3 | ideq 5863 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
| 5 | equcom 2017 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 6 | 4, 5 | bitri 275 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥) |
| 7 | 6 | exbii 1848 | . . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥) |
| 8 | 2, 7 | mpbir 231 | . . 3 ⊢ ∃𝑦 𝑥 I 𝑦 |
| 9 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 10 | 9 | eldm 5911 | . . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦) |
| 11 | 8, 10 | mpbir 231 | . 2 ⊢ 𝑥 ∈ dom I |
| 12 | 1, 11 | mpgbir 1799 | 1 ⊢ dom I = V |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 I cid 5577 dom cdm 5685 |
| 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-pr 5432 |
| 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-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-dm 5695 |
| This theorem is referenced by: dmv 5933 dmresi 6070 idfn 6696 iprc 7933 |
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