Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eleqvrels2 Structured version   Visualization version   GIF version

Theorem eleqvrels2 37975
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 37971 . 2 EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
2 dmeq 5897 . . . . 5 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
32reseq2d 5975 . . . 4 (𝑟 = 𝑅 → ( I ↾ dom 𝑟) = ( I ↾ dom 𝑅))
4 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
53, 4sseq12d 4010 . . 3 (𝑟 = 𝑅 → (( I ↾ dom 𝑟) ⊆ 𝑟 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
6 cnveq 5867 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
76, 4sseq12d 4010 . . 3 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
8 coideq 37626 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
98, 4sseq12d 4010 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
105, 7, 93anbi123d 1432 . 2 (𝑟 = 𝑅 → ((( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅)))
111, 10rabeqel 37635 1 (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wss 3943   I cid 5566  ccnv 5668  dom cdm 5669  cres 5671  ccom 5673   Rels crels 37558   EqvRels ceqvrels 37572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-rels 37868  df-ssr 37881  df-refs 37893  df-refrels 37894  df-syms 37925  df-symrels 37926  df-trs 37955  df-trrels 37956  df-eqvrels 37967
This theorem is referenced by:  eleqvrelsrel  37977
  Copyright terms: Public domain W3C validator