Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elcnvrefrels2 Structured version   Visualization version   GIF version

Theorem elcnvrefrels2 38136
Description: Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.)
Assertion
Ref Expression
elcnvrefrels2 (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))

Proof of Theorem elcnvrefrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfcnvrefrels2 38130 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
2 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
3 dmeq 5906 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5938 . . . . 5 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4xpeq12d 5709 . . . 4 (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
65ineq2d 4210 . . 3 (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
72, 6sseq12d 4010 . 2 (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
81, 7rabeqel 37856 1 (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  cin 3943  wss 3944   I cid 5575   × cxp 5676  dom cdm 5678  ran crn 5679   Rels crels 37781   CnvRefRels ccnvrefrels 37787
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-rels 38087  df-ssr 38100  df-cnvrefs 38127  df-cnvrefrels 38128
This theorem is referenced by:  elcnvrefrelsrel  38138  cosselcnvrefrels2  38140
  Copyright terms: Public domain W3C validator