Theorem refrelcoss3 39751

Description: The class of cosets by 𝑅 is reflexive, see dfrefrel3 39794. (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 39750	. . . 4 ⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
2		idinxpssinxp4 39524	. . . 4 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥)
3	1, 2	mpbir 234	. . 3 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)
4		rncossdmcoss 39743	. . . . 5 ⊢ ran ≀ 𝑅 = dom ≀ 𝑅
5	4	raleqi 3321	. . . 4 ⊢ (∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ↔ ∀𝑦 ∈ dom ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦))
6	5	ralbii 3112	. . 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 234	. 2 ⊢ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦)
8		relcoss 39711	. 2 ⊢ Rel ≀ 𝑅
9	7, 8	pm3.2i 476	1 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wral 3080   class class class wbr 5108  dom cdm 5658  ran crn 5659  Rel wrel 5663   ≀ ccoss 39381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-coss 39699

This theorem is used by:  refrelcoss2  39752