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Theorem eldmqsres2 36045
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 36044 . 2 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅)))
2 df-rex 3059 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
3 19.41v 1957 . . . 4 (∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅) ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
42, 3bitri 278 . . 3 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
54rexbii 3161 . 2 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅 ↔ ∃𝑢𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑢]𝑅))
61, 5bitr4di 292 1 (𝐵𝑉 → (𝐵 ∈ (dom (𝑅𝐴) / (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  wrex 3054  dom cdm 5525  cres 5527  [cec 8318   / cqs 8319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ec 8322  df-qs 8326
This theorem is referenced by:  releldmqs  36393
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