Theorem istsr 18542

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
istsr	⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))

Proof of Theorem istsr

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5867	. . . . 5 ⊢ (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2		istsr.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
3	1, 2	eqtr4di 2782	. . . 4 ⊢ (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
4	3	sqxpeqd 5670	. . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5		id 22	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6		cnveq 5837	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
7	5, 6	uneq12d 4132	. . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅))
8	4, 7	sseq12d 3980	. 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))
9		df-tsr 18526	. 2 ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
10	8, 9	elrab2 3662	1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∪ cun 3912   ⊆ wss 3914   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  PosetRelcps 18523   TosetRel ctsr 18524

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-tsr 18526

This theorem is referenced by:  istsr2  18543  tsrlemax  18545  tsrps  18546  cnvtsr  18547  letsr  18552  tsrdir  18563