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Theorem dfcnvrefrels2 38509
Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38494. (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 38507 . 2 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 38506 . 2 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 dmexg 7923 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
43elv 3482 . . . . 5 dom 𝑟 ∈ V
5 rnexg 7924 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
65elv 3482 . . . . 5 ran 𝑟 ∈ V
74, 6xpex 7771 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
8 inex2g 5325 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9 brcnvssr 38487 . . . 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 38471 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
1211elv 3482 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1312biimpi 216 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1413sseq1d 4026 . . 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 38234 1 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  cin 3961  wss 3962   class class class wbr 5147   I cid 5581   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689   Rels crels 38163   S cssr 38164   CnvRefs ccnvrefs 38168   CnvRefRels ccnvrefrels 38169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-rels 38466  df-ssr 38479  df-cnvrefs 38506  df-cnvrefrels 38507
This theorem is referenced by:  elcnvrefrels2  38515
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