|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > psrn | Structured version Visualization version GIF version | ||
| Description: The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.) | 
| Ref | Expression | 
|---|---|
| psref.1 | ⊢ 𝑋 = dom 𝑅 | 
| Ref | Expression | 
|---|---|
| psrn | ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | psref.1 | . 2 ⊢ 𝑋 = dom 𝑅 | |
| 2 | psdmrn 18618 | . . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) | |
| 3 | eqtr3 2763 | . . 3 ⊢ ((dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅) → dom 𝑅 = ran 𝑅) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅) | 
| 5 | 1, 4 | eqtrid 2789 | 1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 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: cnvtsr 18633 ordtbas2 23199 ordtcnv 23209 ordtrest2 23212 | 
| Copyright terms: Public domain | W3C validator |