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Theorem refrelcoss3 37990
Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 38043. (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 37989 . . . 4 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
2 idinxpssinxp4 37847 . . . 4 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
31, 2mpbir 230 . . 3 𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)
4 rncossdmcoss 37982 . . . . 5 ran ≀ 𝑅 = dom ≀ 𝑅
54raleqi 3313 . . . 4 (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
65ralbii 3083 . . 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 37950 . 2 Rel ≀ 𝑅
97, 8pm3.2i 469 1 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wral 3051   class class class wbr 5143  dom cdm 5672  ran crn 5673  Rel wrel 5677  ccoss 37704
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-coss 37938
This theorem is referenced by:  refrelcoss2  37991
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