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Theorem eldmqsres 35424
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 8337 . 2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴)))
2 eldmres2 35413 . . . . . 6 (𝑢 ∈ V → (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
32elv 3497 . . . . 5 (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))
43anbi1i 623 . . . 4 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)))
5 ecres2 35417 . . . . . . . 8 (𝑢𝐴 → [𝑢](𝑅𝐴) = [𝑢]𝑅)
65eqeq2d 2829 . . . . . . 7 (𝑢𝐴 → (𝐵 = [𝑢](𝑅𝐴) ↔ 𝐵 = [𝑢]𝑅))
76pm5.32i 575 . . . . . 6 ((𝑢𝐴𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴𝐵 = [𝑢]𝑅))
87anbi2i 622 . . . . 5 ((∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢]𝑅)))
9 an21 640 . . . . 5 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))))
10 an12 641 . . . . 5 ((𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢]𝑅)))
118, 9, 103bitr4i 304 . . . 4 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
124, 11bitri 276 . . 3 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
1312rexbii2 3242 . 2 (∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
141, 13syl6bb 288 1 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  Vcvv 3492  dom cdm 5548  cres 5550  [cec 8276   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-qs 8284
This theorem is referenced by:  eldmqsres2  35425
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