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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 4099 | . . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
2 | dmrnssfld 5806 | . . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
3 | 1, 2 | sstri 3924 | . . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | pslem 17808 | . . . . . 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 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7, 7 | breldm 5741 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
9 | 6, 8 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅)) |
10 | 9 | ssrdv 3921 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
11 | 4, 10 | eqssd 3932 | . 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅) |
12 | ssun2 4100 | . . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
13 | 12, 2 | sstri 3924 | . . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
14 | 13 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
15 | 7, 7 | brelrn 5776 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅) |
16 | 6, 15 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅)) |
17 | 16 | ssrdv 3921 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅) |
18 | 14, 17 | eqssd 3932 | . 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅) |
19 | 11, 18 | jca 515 | 1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ran crn 5520 PosetRelcps 17800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ps 17802 |
This theorem is referenced by: psref 17810 psrn 17811 psss 17816 tsrdir 17840 |
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