Theorem psref2 18468

Description: A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)

Assertion

Ref	Expression
psref2	⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))

Proof of Theorem psref2

Step	Hyp	Ref	Expression
1		isps 18466	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
2	1	ibi 267	. 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
3	2	simp3d 1144	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110   ∩ cin 3899   ⊆ wss 3900  ∪ cuni 4857   I cid 5508  ^◡ccnv 5613   ↾ cres 5616   ∘ ccom 5618  Rel wrel 5619  PosetRelcps 18462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3394  df-v 3436  df-in 3907  df-ss 3917  df-uni 4858  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-res 5626  df-ps 18464

This theorem is referenced by:  pslem  18470  cnvps  18476  tsrdir  18502