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Mirrors > Home > MPE Home > Th. List > psdmrn | Structured version Visualization version GIF version |
Description: The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.) |
Ref | Expression |
---|---|
psdmrn | ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3738 | . . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
2 | dmrnssfld 5305 | . . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
3 | 1, 2 | sstri 3577 | . . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | pslem 17029 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥))) | |
6 | 5 | simp2d 1067 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥)) |
7 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7, 7 | breldm 5251 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
9 | 6, 8 | syl6 34 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅)) |
10 | 9 | ssrdv 3574 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
11 | 4, 10 | eqssd 3585 | . 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅) |
12 | ssun2 3739 | . . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
13 | 12, 2 | sstri 3577 | . . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
14 | 13 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
15 | 7, 7 | brelrn 5277 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅) |
16 | 6, 15 | syl6 34 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅)) |
17 | 16 | ssrdv 3574 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅) |
18 | 14, 17 | eqssd 3585 | . 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅) |
19 | 11, 18 | jca 553 | 1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 dom cdm 5038 ran crn 5039 PosetRelcps 17021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ps 17023 |
This theorem is referenced by: psref 17031 psrn 17032 psss 17037 tsrdir 17061 |
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