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Theorem eldmqsres 38269
Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.)
Assertion
Ref Expression
eldmqsres (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [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
StepHypRef Expression
1 elqsg 8807 . 2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴)))
2 eldmres2 38257 . . . . . 6 (𝑢 ∈ V → (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
32elv 3483 . . . . 5 (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))
43anbi1i 624 . . . 4 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)))
5 ecres2 38261 . . . . . . . 8 (𝑢𝐴 → [𝑢](𝑅𝐴) = [𝑢]𝑅)
65eqeq2d 2746 . . . . . . 7 (𝑢𝐴 → (𝐵 = [𝑢](𝑅𝐴) ↔ 𝐵 = [𝑢]𝑅))
76pm5.32i 574 . . . . . 6 ((𝑢𝐴𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴𝐵 = [𝑢]𝑅))
87anbi2i 623 . . . . 5 ((∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢]𝑅)))
9 an21 644 . . . . 5 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))))
10 an12 645 . . . . 5 ((𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢]𝑅)))
118, 9, 103bitr4i 303 . . . 4 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
124, 11bitri 275 . . 3 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
1312rexbii2 3088 . 2 (∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
141, 13bitrdi 287 1 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  dom cdm 5689  cres 5691  [cec 8742   / cqs 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750
This theorem is referenced by:  eldmqsres2  38270
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