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Theorem istsr 17424
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 5462 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2 istsr.1 . . . . 5 𝑋 = dom 𝑅
31, 2syl6eqr 2822 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
43sqxpeqd 5281 . . 3 (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
6 cnveq 5434 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6uneq12d 3917 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
84, 7sseq12d 3781 . 2 (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
9 df-tsr 17408 . 2 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
108, 9elrab2 3516 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wcel 2144  cun 3719  wss 3721   × cxp 5247  ccnv 5248  dom cdm 5249  PosetRelcps 17405   TosetRel ctsr 17406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-cnv 5257  df-dm 5259  df-tsr 17408
This theorem is referenced by:  istsr2  17425  tsrlemax  17427  tsrps  17428  cnvtsr  17429  letsr  17434  tsrdir  17445
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