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Theorem eldmqsres 36185
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 8471 . 2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴)))
2 eldmres2 36174 . . . . . 6 (𝑢 ∈ V → (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
32elv 3427 . . . . 5 (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))
43anbi1i 627 . . . 4 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)))
5 ecres2 36178 . . . . . . . 8 (𝑢𝐴 → [𝑢](𝑅𝐴) = [𝑢]𝑅)
65eqeq2d 2749 . . . . . . 7 (𝑢𝐴 → (𝐵 = [𝑢](𝑅𝐴) ↔ 𝐵 = [𝑢]𝑅))
76pm5.32i 578 . . . . . 6 ((𝑢𝐴𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴𝐵 = [𝑢]𝑅))
87anbi2i 626 . . . . 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 306 . . . 4 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
124, 11bitri 278 . . 3 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
1312rexbii2 3174 . 2 (∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
141, 13bitrdi 290 1 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2111  wrex 3063  Vcvv 3421  dom cdm 5566  cres 5568  [cec 8410   / cqs 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-xp 5572  df-rel 5573  df-cnv 5574  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-ec 8414  df-qs 8418
This theorem is referenced by:  eldmqsres2  36186
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