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Theorem elcnvrefrels2 36244
 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 36240 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
2 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
3 dmeq 5749 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
4 rneq 5782 . . . . 5 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
53, 4xpeq12d 5559 . . . 4 (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
65ineq2d 4119 . . 3 (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
72, 6sseq12d 3927 . 2 (𝑟 = 𝑅 → (𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))))
81, 7rabeqel 35990 1 (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∩ cin 3859   ⊆ wss 3860   I cid 5433   × cxp 5526  dom cdm 5528  ran crn 5529   Rels crels 35929   CnvRefRels ccnvrefrels 35935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-rels 36199  df-ssr 36212  df-cnvrefs 36237  df-cnvrefrels 36238 This theorem is referenced by:  elcnvrefrelsrel  36246  cosselcnvrefrels2  36248
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