Theorem eceldmqs 8788

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 6020	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2		eqid 2760	. 2 ⊢ dom 𝑅 = dom 𝑅
3		ecelqsdmb 8787	. 2 ⊢ (((𝑅 ↾ dom 𝑅) ∈ V ∧ dom 𝑅 = dom 𝑅) → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
4	1, 2, 3	sylancl 598	1 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3450  dom cdm 5655   ↾ cres 5657  [cec 8695   / cqs 8696

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7737

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8699  df-qs 8703

This theorem is used by:  raldmqseu  39114  eceldmqsxrncnvepres  39185  eceldmqsxrncnvepres2  39186  eldisjdmqsim2  39565  eldisjdmqsim  39566  rnqmapeleldisjsim  39611