Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcnvrefrels2 Structured version   Visualization version   GIF version

Theorem dfcnvrefrels2 38529
Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 21-Jul-2021.)
Assertion
Ref Expression
dfcnvrefrels2 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Proof of Theorem dfcnvrefrels2
StepHypRef Expression
1 df-cnvrefrels 38527 . 2 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 38526 . 2 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 dmexg 7923 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
43elv 3485 . . . . 5 dom 𝑟 ∈ V
5 rnexg 7924 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
65elv 3485 . . . . 5 ran 𝑟 ∈ V
74, 6xpex 7773 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
8 inex2g 5320 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9 brcnvssr 38507 . . . 4 (( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
107, 8, 9mp2b 10 . . 3 (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))
11 elrels6 38491 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
1211elv 3485 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1312biimpi 216 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1413sseq1d 4015 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
1510, 14bitrid 283 . 2 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
161, 2, 15abeqinbi 38254 1 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  wss 3951   class class class wbr 5143   I cid 5577   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686   Rels crels 38184   S cssr 38185   CnvRefs ccnvrefs 38189   CnvRefRels ccnvrefrels 38190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-rels 38486  df-ssr 38499  df-cnvrefs 38526  df-cnvrefrels 38527
This theorem is referenced by:  elcnvrefrels2  38535
  Copyright terms: Public domain W3C validator