Theorem psrn 18560

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 18558	. . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3		eqtr3 2754	. . 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 2780	1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∪ cuni 4903  dom cdm 5672  ran crn 5673  PosetRelcps 18549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ps 18551

This theorem is referenced by:  cnvtsr  18573  ordtbas2  23088  ordtcnv  23098  ordtrest2  23101