Theorem psref2 18533

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 18531	. . 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 1143	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∩ cin 3947   ⊆ wss 3948  ∪ cuni 4908   I cid 5573  ^◡ccnv 5675   ↾ cres 5678   ∘ ccom 5680  Rel wrel 5681  PosetRelcps 18527

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-res 5688  df-ps 18529

This theorem is referenced by:  pslem  18535  cnvps  18541  tsrdir  18567