Theorem istsr 18535

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 5893	. . . . 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 5698	. . 3 ⊢ (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5		id 22	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6		cnveq 5863	. . . 4 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
7	5, 6	uneq12d 4156	. . 3 ⊢ (𝑟 = 𝑅 → (𝑟 ∪ ◡𝑟) = (𝑅 ∪ ◡𝑅))
8	4, 7	sseq12d 4007	. 2 ⊢ (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))
9		df-tsr 18519	. 2 ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
10	8, 9	elrab2 3678	1 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ∪ cun 3938   ⊆ wss 3940   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  PosetRelcps 18516   TosetRel ctsr 18517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-dm 5676  df-tsr 18519

This theorem is referenced by:  istsr2  18536  tsrlemax  18538  tsrps  18539  cnvtsr  18540  letsr  18545  tsrdir  18556