Theorem refrelcoss2 39752

Description: The class of cosets by 𝑅 is reflexive, see dfrefrel2 39793. (Contributed by Peter Mazsa, 30-Jul-2019.)

Assertion

Ref	Expression
refrelcoss2	⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Proof of Theorem refrelcoss2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		refrelcoss3 39751	. 2 ⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)
2		idinxpss 39516	. . 3 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ↔ ∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦))
3	2	anbi1i 636	. 2 ⊢ ((( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) ↔ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅))
4	1, 3	mpbir 234	1 ⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wral 3080   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5108   I cid 5552   × cxp 5656  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:  cossssid  39755  refrelcoss  39801