Theorem eldmqsres 39028

Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.)

Assertion

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

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

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥, 𝑢)

Proof of Theorem eldmqsres

Step	Hyp	Ref	Expression
1		elqsg 8766	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴)))
2		eldmres2 39017	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
3	2	elv 3458	. . . . 5 ⊢ (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))
4	3	anbi1i 636	. . . 4 ⊢ ((𝑢 ∈ dom (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ ((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)))
5		elecreseq 8749	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → [𝑢](𝑅 ↾ 𝐴) = [𝑢]𝑅)
6	5	eqeq2d 2773	. . . . . . 7 ⊢ (𝑢 ∈ 𝐴 → (𝐵 = [𝑢](𝑅 ↾ 𝐴) ↔ 𝐵 = [𝑢]𝑅))
7	6	pm5.32i 585	. . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅))
8	7	anbi2i 635	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴))) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅)))
9		an21 657	. . . . 5 ⊢ (((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴))))
10		an12 658	. . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢 ∈ 𝐴 ∧ 𝐵 = [𝑢]𝑅)))
11	8, 9, 10	3bitr4i 306	. . . 4 ⊢ (((𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
12	4, 11	bitri 278	. . 3 ⊢ ((𝑢 ∈ dom (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑢](𝑅 ↾ 𝐴)) ↔ (𝑢 ∈ 𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))
13	12	rexbii2 3107	. 2 ⊢ (∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))
14	1, 13	bitrdi 290	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453  dom cdm 5659   ↾ cres 5661  [cec 8697   / cqs 8698

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 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8701  df-qs 8705

This theorem is used by:  eldmqsres2  39029