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

Theorem cnvtsr 17423
Description: The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
cnvtsr (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )

Proof of Theorem cnvtsr
StepHypRef Expression
1 tsrps 17422 . . 3 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2 cnvps 17413 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . 2 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
4 eqid 2760 . . . . 5 dom 𝑅 = dom 𝑅
54istsr 17418 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
65simprbi 483 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅))
74psrn 17410 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
81, 7syl 17 . . . 4 (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
98sqxpeqd 5298 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10 psrel 17404 . . . . . . 7 (𝑅 ∈ PosetRel → Rel 𝑅)
111, 10syl 17 . . . . . 6 (𝑅 ∈ TosetRel → Rel 𝑅)
12 dfrel2 5741 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
1311, 12sylib 208 . . . . 5 (𝑅 ∈ TosetRel → 𝑅 = 𝑅)
1413uneq2d 3910 . . . 4 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
15 uncom 3900 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1614, 15syl6req 2811 . . 3 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
176, 9, 163sstr3d 3788 . 2 (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅))
18 df-rn 5277 . . 3 ran 𝑅 = dom 𝑅
1918istsr 17418 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅)))
203, 17, 19sylanbrc 701 1 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cun 3713  wss 3715   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  Rel wrel 5271  PosetRelcps 17399   TosetRel ctsr 17400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ps 17401  df-tsr 17402
This theorem is referenced by:  ordtbas2  21197  ordtrest2  21210  cnvordtrestixx  30268
  Copyright terms: Public domain W3C validator