Theorem istsr2 18533

Description: The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
istsr.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
istsr2	⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦

Proof of Theorem istsr2

Step	Hyp	Ref	Expression
1		istsr.1	. . 3 ⊢ 𝑋 = dom 𝑅
2	1	istsr 18532	. 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))
3		qfto 6119	. . 3 ⊢ ((𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
4	3	anbi2i 624	. 2 ⊢ ((𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)) ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
5	2, 4	bitri 275	1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ∪ cun 3945   ⊆ wss 3947   class class class wbr 5147   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  PosetRelcps 18513   TosetRel ctsr 18514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-tsr 18516

This theorem is referenced by:  tsrlin  18534  tsrss  18538