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Theorem istsr 17040
 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 5246 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2 istsr.1 . . . . 5 𝑋 = dom 𝑅
31, 2syl6eqr 2662 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
43sqxpeqd 5065 . . 3 (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
6 cnveq 5218 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6uneq12d 3730 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
84, 7sseq12d 3597 . 2 (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
9 df-tsr 17024 . 2 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
108, 9elrab2 3333 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540   × cxp 5036  ◡ccnv 5037  dom cdm 5038  PosetRelcps 17021   TosetRel ctsr 17022 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-tsr 17024 This theorem is referenced by:  istsr2  17041  tsrlemax  17043  tsrps  17044  cnvtsr  17045  letsr  17050  tsrdir  17061
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