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Mirrors > Home > ILE Home > Th. List > dmi | 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 3352 | . 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I ) | |
2 | a9ev 1660 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑥 | |
3 | vex 2663 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 4661 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | equcom 1667 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
6 | 4, 5 | bitri 183 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥) |
7 | 6 | exbii 1569 | . . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥) |
8 | 2, 7 | mpbir 145 | . . 3 ⊢ ∃𝑦 𝑥 I 𝑦 |
9 | vex 2663 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | 9 | eldm 4706 | . . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦) |
11 | 8, 10 | mpbir 145 | . 2 ⊢ 𝑥 ∈ dom I |
12 | 1, 11 | mpgbir 1414 | 1 ⊢ dom I = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 class class class wbr 3899 I cid 4180 dom cdm 4509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-dm 4519 |
This theorem is referenced by: dmv 4725 iprc 4777 dmresi 4844 climshft2 11043 |
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