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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 4150 | . . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
2 | dmrnssfld 5843 | . . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
3 | 1, 2 | sstri 3978 | . . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
4 | 3 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | pslem 17818 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥))) | |
6 | 5 | simp2d 1139 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥)) |
7 | vex 3499 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | 7, 7 | breldm 5779 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
9 | 6, 8 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅)) |
10 | 9 | ssrdv 3975 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
11 | 4, 10 | eqssd 3986 | . 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅) |
12 | ssun2 4151 | . . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
13 | 12, 2 | sstri 3978 | . . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
14 | 13 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
15 | 7, 7 | brelrn 5814 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅) |
16 | 6, 15 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅)) |
17 | 16 | ssrdv 3975 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅) |
18 | 14, 17 | eqssd 3986 | . 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅) |
19 | 11, 18 | jca 514 | 1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ran crn 5558 PosetRelcps 17810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ps 17812 |
This theorem is referenced by: psref 17820 psrn 17821 psss 17826 tsrdir 17850 |
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