Theorem psref2 18615

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 18613	. . 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 1145	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907   I cid 5577  ^◡ccnv 5684   ↾ cres 5687   ∘ ccom 5689  Rel wrel 5690  PosetRelcps 18609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-res 5697  df-ps 18611

This theorem is referenced by:  pslem  18617  cnvps  18623  tsrdir  18649