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Theorem eldmqs1cossres 35445
Description: Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.)
Assertion
Ref Expression
eldmqs1cossres (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝑅,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem eldmqs1cossres
StepHypRef Expression
1 elqsg 8205 . . 3 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴)))
2 df-rex 3113 . . . 4 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)))
3 eldm1cossres2 35253 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅))
43elv 3445 . . . . . 6 (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅)
54anbi1i 623 . . . . 5 ((𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
65exbii 1833 . . . 4 (∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
72, 6bitri 276 . . 3 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
81, 7syl6bb 288 . 2 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴))))
9 df-rex 3113 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
109rexbii 3213 . . 3 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
11 rexcom4 3215 . . . 4 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
12 r19.41v 3310 . . . . 5 (∃𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1312exbii 1833 . . . 4 (∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1411, 13bitri 276 . . 3 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1510, 14bitri 276 . 2 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
168, 15syl6bbr 290 1 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wex 1765  wcel 2083  wrex 3108  Vcvv 3440  dom cdm 5450  cres 5452  [cec 8144   / cqs 8145  ccoss 35006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-ec 8148  df-qs 8152  df-coss 35211
This theorem is referenced by:  releldmqscoss  35446
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