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Theorem dfcnvrefrels2 34899
 Description: Alternate definition of the class of converse reflexive relations. Cf. the comment of dfrefrels2 34886. (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 34897 . 2 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 34896 . 2 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 dmexg 7375 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
43elv 3401 . . . . 5 dom 𝑟 ∈ V
5 rnexg 7376 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
65elv 3401 . . . . 5 ran 𝑟 ∈ V
74, 6xpex 7240 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
8 inex2ALTV 34728 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9 brcnvssr 34879 . . . 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 34863 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
1211elv 3401 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1312biimpi 208 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1413sseq1d 3850 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
1510, 14syl5bb 275 . 2 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
161, 2, 15abeqinbi 34648 1 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1601   ∈ wcel 2106  {crab 3093  Vcvv 3397   ∩ cin 3790   ⊆ wss 3791   class class class wbr 4886   I cid 5260   × cxp 5353  ◡ccnv 5354  dom cdm 5355  ran crn 5356   Rels crels 34603   S cssr 34604   CnvRefs ccnvrefs 34608   CnvRefRels ccnvrefrels 34609 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-rels 34858  df-ssr 34871  df-cnvrefs 34896  df-cnvrefrels 34897 This theorem is referenced by:  elcnvrefrels2  34903
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