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Theorem istsr 18544
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) ⊆ (𝑅𝑅)))

Proof of Theorem istsr
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dmeq 5894 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2 istsr.1 . . . . 5 𝑋 = dom 𝑅
31, 2eqtr4di 2782 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
43sqxpeqd 5699 . . 3 (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
6 cnveq 5864 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6uneq12d 4157 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
84, 7sseq12d 4008 . 2 (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
9 df-tsr 18528 . 2 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
108, 9elrab2 3679 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098  cun 3939  wss 3941   × cxp 5665  ccnv 5666  dom cdm 5667  PosetRelcps 18525   TosetRel ctsr 18526
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-tsr 18528
This theorem is referenced by:  istsr2  18545  tsrlemax  18547  tsrps  18548  cnvtsr  18549  letsr  18554  tsrdir  18565
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