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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 4170 | . . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
2 | dmrnssfld 5973 | . . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
3 | 1, 2 | sstri 3986 | . . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | pslem 18567 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥))) | |
6 | 5 | simp2d 1140 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥)) |
7 | vex 3465 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7, 7 | breldm 5911 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
9 | 6, 8 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅)) |
10 | 9 | ssrdv 3982 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
11 | 4, 10 | eqssd 3994 | . 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅) |
12 | ssun2 4171 | . . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
13 | 12, 2 | sstri 3986 | . . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
14 | 13 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
15 | 7, 7 | brelrn 5944 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅) |
16 | 6, 15 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅)) |
17 | 16 | ssrdv 3982 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅) |
18 | 14, 17 | eqssd 3994 | . 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅) |
19 | 11, 18 | jca 510 | 1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ⊆ wss 3944 ∪ cuni 4909 class class class wbr 5149 dom cdm 5678 ran crn 5679 PosetRelcps 18559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ps 18561 |
This theorem is referenced by: psref 18569 psrn 18570 psss 18575 tsrdir 18599 |
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