Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refrelcoss3 Structured version   Visualization version   GIF version

Theorem refrelcoss3 37846
Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 37899. (Contributed by Peter Mazsa, 30-Jul-2019.)
Assertion
Ref Expression
refrelcoss3 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem refrelcoss3
StepHypRef Expression
1 refrelcosslem 37845 . . . 4 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
2 idinxpssinxp4 37702 . . . 4 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
31, 2mpbir 230 . . 3 𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)
4 rncossdmcoss 37838 . . . . 5 ran ≀ 𝑅 = dom ≀ 𝑅
54raleqi 3317 . . . 4 (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
65ralbii 3087 . . 3 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
73, 6mpbir 230 . 2 𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)
8 relcoss 37806 . 2 Rel ≀ 𝑅
97, 8pm3.2i 470 1 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wral 3055   class class class wbr 5141  dom cdm 5669  ran crn 5670  Rel wrel 5674  ccoss 37556
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-10 2129  ax-11 2146  ax-12 2163  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-nf 1778  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-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-coss 37794
This theorem is referenced by:  refrelcoss2  37847
  Copyright terms: Public domain W3C validator