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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 17593 | . . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) | |
3 | eqtr3 2801 | . . 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 | syl5eq 2826 | 1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cuni 4671 dom cdm 5355 ran crn 5356 PosetRelcps 17584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ps 17586 |
This theorem is referenced by: cnvtsr 17608 ordtbas2 21403 ordtcnv 21413 ordtrest2 21416 |
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