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Theorem elrefrels2 34559
Description: Element of the class of all reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
Assertion
Ref Expression
elrefrels2 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))

Proof of Theorem elrefrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfrefrels2 34555 . 2 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
2 dmeq 5467 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
3 rneq 5494 . . . . 5 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
42, 3xpeq12d 5285 . . . 4 (𝑟 = 𝑅 → (dom 𝑟 × ran 𝑟) = (dom 𝑅 × ran 𝑅))
54ineq2d 3945 . . 3 (𝑟 = 𝑅 → ( I ∩ (dom 𝑟 × ran 𝑟)) = ( I ∩ (dom 𝑅 × ran 𝑅)))
6 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
75, 6sseq12d 3763 . 2 (𝑟 = 𝑅 → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 ↔ ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅))
81, 7rabeqel 34312 1 (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1620  wcel 2127  cin 3702  wss 3703   I cid 5161   × cxp 5252  dom cdm 5254  ran crn 5255   Rels crels 34267   RefRels crefrels 34270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-rels 34527  df-ssr 34540  df-refs 34552  df-refrels 34553
This theorem is referenced by:  elrefrelsrel  34561
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