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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 4178 | . . . . 5 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
| 2 | dmrnssfld 5984 | . . . . 5 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
| 3 | 1, 2 | sstri 3993 | . . . 4 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
| 5 | pslem 18617 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥 ∧ 𝑥𝑅𝑥) → 𝑥 = 𝑥))) | |
| 6 | 5 | simp2d 1144 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥𝑅𝑥)) |
| 7 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 8 | 7, 7 | breldm 5919 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ dom 𝑅) |
| 9 | 6, 8 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ dom 𝑅)) |
| 10 | 9 | ssrdv 3989 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
| 11 | 4, 10 | eqssd 4001 | . 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅) |
| 12 | ssun2 4179 | . . . . 5 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
| 13 | 12, 2 | sstri 3993 | . . . 4 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
| 14 | 13 | a1i 11 | . . 3 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
| 15 | 7, 7 | brelrn 5953 | . . . . 5 ⊢ (𝑥𝑅𝑥 → 𝑥 ∈ ran 𝑅) |
| 16 | 6, 15 | syl6 35 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑥 ∈ ∪ ∪ 𝑅 → 𝑥 ∈ ran 𝑅)) |
| 17 | 16 | ssrdv 3989 | . . 3 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 ⊆ ran 𝑅) |
| 18 | 14, 17 | eqssd 4001 | . 2 ⊢ (𝑅 ∈ PosetRel → ran 𝑅 = ∪ ∪ 𝑅) |
| 19 | 11, 18 | jca 511 | 1 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 ran crn 5686 PosetRelcps 18609 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ps 18611 |
| This theorem is referenced by: psref 18619 psrn 18620 psss 18625 tsrdir 18649 |
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