Theorem refrelcoss3 37333

Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 37386. (Contributed by Peter Mazsa, 30-Jul-2019.)

Assertion

Ref	Expression
refrelcoss3	⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem refrelcoss3

Step	Hyp	Ref	Expression
1		refrelcosslem 37332	. . . 4 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
2		idinxpssinxp4 37189	. . . 4 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥)
3	1, 2	mpbir 230	. . 3 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)
4		rncossdmcoss 37325	. . . . 5 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅
5	4	raleqi 3324	. . . 4 ⊢ (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦))
6	5	ralbii 3094	. . 3 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦))
7	3, 6	mpbir 230	. 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)
8		relcoss 37293	. 2 ⊢ Rel ≀ 𝑅
9	7, 8	pm3.2i 472	1 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 397  ∀wral 3062   class class class wbr 5149  dom cdm 5677  ran crn 5678  Rel wrel 5682   ≀ ccoss 37043

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 5300  ax-nul 5307  ax-pr 5428

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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-coss 37281

This theorem is referenced by:  refrelcoss2  37334