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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 17805 | . . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) | |
3 | eqtr3 2840 | . . 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 2865 | 1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 dom cdm 5548 ran crn 5549 PosetRelcps 17796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ps 17798 |
This theorem is referenced by: cnvtsr 17820 ordtbas2 21727 ordtcnv 21737 ordtrest2 21740 |
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