Theorem psrn 18524

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 18522	. . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3		eqtr3 2758	. . 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 2784	1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∪ cuni 4907  dom cdm 5675  ran crn 5676  PosetRelcps 18513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ps 18515

This theorem is referenced by:  cnvtsr  18537  ordtbas2  22686  ordtcnv  22696  ordtrest2  22699