Theorem eceldmqs 8728

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 5987	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2		eqid 2737	. 2 ⊢ dom 𝑅 = dom 𝑅
3		ecelqsdmb 8727	. 2 ⊢ (((𝑅 ↾ dom 𝑅) ∈ V ∧ dom 𝑅 = dom 𝑅) → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
4	1, 2, 3	sylancl 587	1 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3430  dom cdm 5625   ↾ cres 5627  [cec 8635   / cqs 8636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643

This theorem is referenced by:  raldmqseu  38703  eceldmqsxrncnvepres  38774  eceldmqsxrncnvepres2  38775  eldisjdmqsim2  39154  eldisjdmqsim  39155  rnqmapeleldisjsim  39200