Theorem psref2 18585

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 18583	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
2	1	ibi 269	. 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
3	2	simp3d 1156	1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∩ cin 3903   ⊆ wss 3904  ∪ cuni 4864   I cid 5539  ^◡ccnv 5644   ↾ cres 5647   ∘ ccom 5649  Rel wrel 5650  PosetRelcps 18579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-in 3911  df-ss 3921  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-res 5657  df-ps 18581

This theorem is referenced by:  pslem  18587  cnvps  18593  tsrdir  18619