MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istsr Structured version   Visualization version   GIF version

Theorem istsr 17817
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 5771 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
2 istsr.1 . . . . 5 𝑋 = dom 𝑅
31, 2syl6eqr 2879 . . . 4 (𝑟 = 𝑅 → dom 𝑟 = 𝑋)
43sqxpeqd 5586 . . 3 (𝑟 = 𝑅 → (dom 𝑟 × dom 𝑟) = (𝑋 × 𝑋))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
6 cnveq 5743 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6uneq12d 4144 . . 3 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
84, 7sseq12d 4004 . 2 (𝑟 = 𝑅 → ((dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟) ↔ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
9 df-tsr 17801 . 2 TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟𝑟)}
108, 9elrab2 3687 1 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107  cun 3938  wss 3940   × cxp 5552  ccnv 5553  dom cdm 5554  PosetRelcps 17798   TosetRel ctsr 17799
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-cnv 5562  df-dm 5564  df-tsr 17801
This theorem is referenced by:  istsr2  17818  tsrlemax  17820  tsrps  17821  cnvtsr  17822  letsr  17827  tsrdir  17838
  Copyright terms: Public domain W3C validator