Theorem eldmqsres 39417

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 8812	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ dom (𝑅 ↾ 𝐴)𝐵 = [𝑢](𝑅 ↾ 𝐴)))
2		eldmres2 39406	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 ∈ dom (𝑅 ↾ 𝐴) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
3	2	elv 3459	. . . . 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 8795	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐴 → [𝑢](𝑅 ↾ 𝐴) = [𝑢]𝑅)
6	5	eqeq2d 2774	. . . . . . 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 3109	. 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 2147  ∃wrex 3090  Vcvv 3454  dom cdm 5655   ↾ cres 5657  [cec 8743   / cqs 8744

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8747  df-qs 8751

This theorem is used by:  eldmqsres2  39418