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Theorem refrelcoss3 36971
Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 37024. (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 36970 . . . 4 𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥
2 idinxpssinxp4 36827 . . . 4 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥)
31, 2mpbir 230 . . 3 𝑥 ∈ dom ≀ 𝑅𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦)
4 rncossdmcoss 36963 . . . . 5 ran ≀ 𝑅 = dom ≀ 𝑅
54raleqi 3310 . . . 4 (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦))
65ralbii 3093 . . 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 36931 . 2 Rel ≀ 𝑅
97, 8pm3.2i 472 1 (∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wral 3061   class class class wbr 5106  dom cdm 5634  ran crn 5635  Rel wrel 5639  ccoss 36680
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 5257  ax-nul 5264  ax-pr 5385
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-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-coss 36919
This theorem is referenced by:  refrelcoss2  36972
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