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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 17207 | . . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅)) | |
3 | eqtr3 2643 | . . 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 2668 | 1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 dom cdm 5114 ran crn 5115 PosetRelcps 17198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ps 17200 |
This theorem is referenced by: cnvtsr 17222 ordtbas2 20995 ordtcnv 21005 ordtrest2 21008 |
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