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

Theorem eqvreldmqs 37187
Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
Assertion
Ref Expression
eqvreldmqs (( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Proof of Theorem eqvreldmqs
StepHypRef Expression
1 df-coeleqvrel 37099 . . 3 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
21bicomi 223 . 2 ( EqvRel ≀ ( E ↾ 𝐴) ↔ CoElEqvRel 𝐴)
3 dmqs1cosscnvepreseq 37174 . 2 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
42, 3anbi12i 628 1 (( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542   cuni 4869   E cep 5540  ccnv 5636  dom cdm 5637  cres 5639   / cqs 8653  ccoss 36684  ccoels 36685   EqvRel weqvrel 36701   CoElEqvRel wcoeleqvrel 36703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-eprel 5541  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ec 8656  df-qs 8660  df-coss 36923  df-coels 36924  df-coeleqvrel 37099
This theorem is referenced by:  mpet3  37348
  Copyright terms: Public domain W3C validator