Theorem eldmqsres2 38889

Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.)

Assertion

Ref	Expression
eldmqsres2	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))

Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝑅,𝑥

Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem eldmqsres2

Step	Hyp	Ref	Expression
1		eldmqsres 38888	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
2		df-rex 3088	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
3		19.41v 1977	. . . 4 ⊢ (∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
4	2, 3	bitri 278	. . 3 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
5	4	rexbii 3110	. 2 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
6	1, 5	bitr4di 292	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∃wrex 3087  dom cdm 5661   ↾ cres 5663  [cec 8691   / cqs 8692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8695  df-qs 8699

This theorem is referenced by:  releldmqs  39338