Theorem eceldmqs 8724

Description: 𝑅-coset in its domain quotient. This is the bridge between 𝐴 in the domain and its block [𝐴]𝑅 in its domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) (Revised by Peter Mazsa, 22-Nov-2025.)

Assertion

Ref	Expression
eceldmqs	⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))

Proof of Theorem eceldmqs

Step	Hyp	Ref	Expression
1		resexg 5979	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2		eqid 2739	. 2 ⊢ dom 𝑅 = dom 𝑅
3		ecelqsdmb 8723	. 2 ⊢ (((𝑅 ↾ dom 𝑅) ∈ V ∧ dom 𝑅 = dom 𝑅) → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
4	1, 2, 3	sylancl 592	1 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  Vcvv 3431  dom cdm 5618   ↾ cres 5620  [cec 8631   / cqs 8632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635  df-qs 8639

This theorem is referenced by:  raldmqseu  38732  eceldmqsxrncnvepres  38803  eceldmqsxrncnvepres2  38804  eldisjdmqsim2  39183  eldisjdmqsim  39184  rnqmapeleldisjsim  39229