Theorem refrelcoss3 36581

Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 36634. (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 36580	. . . 4 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
2		idinxpssinxp4 36455	. . . 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 36573	. . . . 5 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅
5	4	raleqi 3346	. . . 4 ⊢ (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦))
6	5	ralbii 3092	. . 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 36546	. 2 ⊢ Rel ≀ 𝑅
9	7, 8	pm3.2i 471	1 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 396  ∀wral 3064   class class class wbr 5074  dom cdm 5589  ran crn 5590  Rel wrel 5594   ≀ ccoss 36333

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-coss 36537

This theorem is referenced by:  refrelcoss2  36582