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Theorem psrn 17430
 Description: The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
Hypothesis
Ref Expression
psref.1 𝑋 = dom 𝑅
Assertion
Ref Expression
psrn (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)

Proof of Theorem psrn
StepHypRef Expression
1 psref.1 . 2 𝑋 = dom 𝑅
2 psdmrn 17428 . . 3 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
3 eqtr3 2781 . . 3 ((dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅) → dom 𝑅 = ran 𝑅)
42, 3syl 17 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
51, 4syl5eq 2806 1 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∪ cuni 4588  dom cdm 5266  ran crn 5267  PosetRelcps 17419 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ps 17421 This theorem is referenced by:  cnvtsr  17443  ordtbas2  21217  ordtcnv  21227  ordtrest2  21230
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