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Theorem istsr 17211
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 5322 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2 istsr.1 . . . . 5 𝑋 = dom 𝑅
31, 2syl6eqr 2673 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
43sqxpeqd 5139 . . 3 (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
6 cnveq 5294 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6uneq12d 3766 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
84, 7sseq12d 3632 . 2 (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
9 df-tsr 17195 . 2 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
108, 9elrab2 3364 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1482  wcel 1989  cun 3570  wss 3572   × cxp 5110  ccnv 5111  dom cdm 5112  PosetRelcps 17192   TosetRel ctsr 17193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-tsr 17195
This theorem is referenced by:  istsr2  17212  tsrlemax  17214  tsrps  17215  cnvtsr  17216  letsr  17221  tsrdir  17232
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