Theorem psrn 17818

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 17816	. . 3 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3		eqtr3 2843	. . 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	syl5eq 2868	1 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∪ cuni 4837  dom cdm 5554  ran crn 5555  PosetRelcps 17807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ps 17809

This theorem is referenced by:  cnvtsr  17831  ordtbas2  21798  ordtcnv  21808  ordtrest2  21811