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Theorem eldmqsres 34375
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 7965 . 2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴)))
2 eldmres2 34362 . . . . . 6 (𝑢 ∈ V → (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅)))
32elv 34309 . . . . 5 (𝑢 ∈ dom (𝑅𝐴) ↔ (𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅))
43anbi1i 733 . . . 4 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)))
5 ecres2 34368 . . . . . . . 8 (𝑢𝐴 → [𝑢](𝑅𝐴) = [𝑢]𝑅)
65eqeq2d 2770 . . . . . . 7 (𝑢𝐴 → (𝐵 = [𝑢](𝑅𝐴) ↔ 𝐵 = [𝑢]𝑅))
76pm5.32i 672 . . . . . 6 ((𝑢𝐴𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴𝐵 = [𝑢]𝑅))
87anbi2i 732 . . . . 5 ((∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢]𝑅)))
9 3ancoma 1084 . . . . . 6 ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢](𝑅𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝑢𝐴𝐵 = [𝑢](𝑅𝐴)))
10 df-3an 1074 . . . . . 6 ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢](𝑅𝐴)) ↔ ((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)))
11 3anass 1081 . . . . . 6 ((∃𝑥 𝑥 ∈ [𝑢]𝑅𝑢𝐴𝐵 = [𝑢](𝑅𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))))
129, 10, 113bitr3i 290 . . . . 5 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢](𝑅𝐴))))
13 an12 873 . . . . 5 ((𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ (𝑢𝐴𝐵 = [𝑢]𝑅)))
148, 12, 133bitr4i 292 . . . 4 (((𝑢𝐴 ∧ ∃𝑥 𝑥 ∈ [𝑢]𝑅) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
154, 14bitri 264 . . 3 ((𝑢 ∈ dom (𝑅𝐴) ∧ 𝐵 = [𝑢](𝑅𝐴)) ↔ (𝑢𝐴 ∧ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
1615rexbii2 3177 . 2 (∃𝑢 ∈ dom (𝑅𝐴)𝐵 = [𝑢](𝑅𝐴) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
171, 16syl6bb 276 1 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wrex 3051  Vcvv 3340  dom cdm 5266  cres 5268  [cec 7909   / cqs 7910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ec 7913  df-qs 7917
This theorem is referenced by:  eldmqsres2  34376
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