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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 3391 | . 2 ⊢ (dom I = V ↔ ∀𝑥 𝑥 ∈ dom I ) | |
2 | ax6ev 2074 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑥 | |
3 | vex 3388 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 5478 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | equcom 2117 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
6 | 4, 5 | bitri 267 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑦 = 𝑥) |
7 | 6 | exbii 1944 | . . . 4 ⊢ (∃𝑦 𝑥 I 𝑦 ↔ ∃𝑦 𝑦 = 𝑥) |
8 | 2, 7 | mpbir 223 | . . 3 ⊢ ∃𝑦 𝑥 I 𝑦 |
9 | vex 3388 | . . . 4 ⊢ 𝑥 ∈ V | |
10 | 9 | eldm 5524 | . . 3 ⊢ (𝑥 ∈ dom I ↔ ∃𝑦 𝑥 I 𝑦) |
11 | 8, 10 | mpbir 223 | . 2 ⊢ 𝑥 ∈ dom I |
12 | 1, 11 | mpgbir 1895 | 1 ⊢ dom I = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∃wex 1875 ∈ wcel 2157 Vcvv 3385 class class class wbr 4843 I cid 5219 dom cdm 5312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-dm 5322 |
This theorem is referenced by: dmv 5544 dmresi 5676 iprc 7336 |
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