Theorem refrelcoss3 38459

Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 38512. (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 38458	. . . 4 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
2		idinxpssinxp4 38316	. . . 4 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥)
3	1, 2	mpbir 231	. . 3 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)
4		rncossdmcoss 38451	. . . . 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 3093	. . 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 231	. 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)
8		relcoss 38419	. 2 ⊢ Rel ≀ 𝑅
9	7, 8	pm3.2i 470	1 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395  ∀wral 3061   class class class wbr 5151  dom cdm 5693  ran crn 5694  Rel wrel 5698   ≀ ccoss 38176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-coss 38407

This theorem is referenced by:  refrelcoss2  38460