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Theorem dfcnvrefrels2 34599
Description: Alternate definition of the class of all converse reflexive relations. Cf. the comment of dfrefrels2 34586. (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 34597 . 2 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 34596 . 2 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 dmexg 7262 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
43elv 34309 . . . . 5 dom 𝑟 ∈ V
5 rnexg 7263 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
65elv 34309 . . . . 5 ran 𝑟 ∈ V
74, 6xpex 7127 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
8 inex2ALTV 34429 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9 brcnvssr 34579 . . . 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 34563 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
1211elv 34309 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1312biimpi 206 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1413sseq1d 3773 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
1510, 14syl5bb 272 . 2 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
161, 2, 15abeqinbi 34342 1 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cin 3714  wss 3715   class class class wbr 4804   I cid 5173   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267   Rels crels 34298   S cssr 34299   CnvRefs ccnvrefs 34303   CnvRefRels ccnvrefrels 34304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-rels 34558  df-ssr 34571  df-cnvrefs 34596  df-cnvrefrels 34597
This theorem is referenced by:  elcnvrefrels2  34603
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