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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 3302 | . 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I ) | |
2 | a9ev 1632 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑥 | |
3 | vex 2622 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 4588 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | equcom 1639 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
6 | 4, 5 | bitri 182 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥) |
7 | 6 | exbii 1541 | . . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥) |
8 | 2, 7 | mpbir 144 | . . 3 ⊢ ∃𝑦 𝑥 I 𝑦 |
9 | vex 2622 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | 9 | eldm 4633 | . . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦) |
11 | 8, 10 | mpbir 144 | . 2 ⊢ 𝑥 ∈ dom I |
12 | 1, 11 | mpgbir 1387 | 1 ⊢ dom I = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∃wex 1426 ∈ wcel 1438 Vcvv 2619 class class class wbr 3845 I cid 4115 dom cdm 4438 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-dm 4448 |
This theorem is referenced by: dmv 4652 iprc 4701 dmresi 4767 climshft2 10691 |
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