Theorem psrn 18620

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

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)

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