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Theorem eldmqsres2 34376
Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.)
Assertion
Ref Expression
eldmqsres2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝑅,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem eldmqsres2
StepHypRef Expression
1 eldmqsres 34375 . 2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
2 df-rex 3056 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
3 19.41v 2026 . . . 4 (∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
42, 3bitri 264 . . 3 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
54rexbii 3179 . 2 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
61, 5syl6bbr 278 1 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wrex 3051  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: (None)
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