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

Theorem refrelcoss3 34756
 Description: The class of cosets by 𝑅 is reflexive, cf. dfrefrel3 34809. (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 34755 . . . 4 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
2 idinxpssinxp4 34634 . . . 4 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
31, 2mpbir 223 . . 3 𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)
4 rncossdmcoss 34748 . . . . 5 ran ≀ 𝑅 = dom ≀ 𝑅
54raleqi 3354 . . . 4 (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
65ralbii 3189 . . 3 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
73, 6mpbir 223 . 2 𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)
8 relcoss 34721 . 2 Rel ≀ 𝑅
97, 8pm3.2i 464 1 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656  ∀wral 3117   class class class wbr 4875  dom cdm 5346  ran crn 5347  Rel wrel 5351   ≀ ccoss 34519 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-coss 34712 This theorem is referenced by:  refrelcoss2  34757
 Copyright terms: Public domain W3C validator