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 35646
Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 35633. (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 35644 . 2 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 35643 . 2 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 dmexg 7602 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
43elv 3497 . . . . 5 dom 𝑟 ∈ V
5 rnexg 7603 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
65elv 3497 . . . . 5 ran 𝑟 ∈ V
74, 6xpex 7465 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
8 inex2g 5215 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9 brcnvssr 35626 . . . 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 35610 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
1211elv 3497 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1312biimpi 217 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1413sseq1d 3995 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
1510, 14syl5bb 284 . 2 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
161, 2, 15abeqinbi 35396 1 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cin 3932  wss 3933   class class class wbr 5057   I cid 5452   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549   Rels crels 35336   S cssr 35337   CnvRefs ccnvrefs 35341   CnvRefRels ccnvrefrels 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-rels 35605  df-ssr 35618  df-cnvrefs 35643  df-cnvrefrels 35644
This theorem is referenced by:  elcnvrefrels2  35650
  Copyright terms: Public domain W3C validator