Theorem psrn 18799

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 18797	. . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3		eqtr3 2785	. . 3 ⊢ ((dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅) → dom 𝑅 = ran 𝑅)
4	2, 3	syl 18	. 2 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
5	1, 4	eqtrid 2810	1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∪ cuni 4872  dom cdm 5658  ran crn 5659  PosetRelcps 18788

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ps 18790

This theorem is used by:  cnvtsr  18812  ordtbas2  23564  ordtcnv  23574  ordtrest2  23577