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Theorem eleqvrels2 34830
Description: Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
Assertion
Ref Expression
eleqvrels2 (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Proof of Theorem eleqvrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfeqvrels2 34826 . 2 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
2 dmeq 5527 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32reseq2d 5600 . . . 4 (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
53, 4sseq12d 3830 . . 3 (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6 cnveq 5499 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
76, 4sseq12d 3830 . . 3 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
8 coideq 34510 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
98, 4sseq12d 3830 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
105, 7, 93anbi123d 1561 . 2 (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
111, 10rabeqel 34519 1 (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wss 3769   I cid 5219  ccnv 5311  dom cdm 5312  cres 5314  ccom 5316   Rels crels 34471   EqvRels ceqvrels 34485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-rels 34729  df-ssr 34742  df-refs 34754  df-refrels 34755  df-syms 34782  df-symrels 34783  df-trs 34812  df-trrels 34813  df-eqvrels 34823
This theorem is referenced by:  eleqvrelsrel  34832
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