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Theorem eldmqs1cossres 37121
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 8707 . . 3 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴)))
2 df-rex 3074 . . . 4 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)))
3 eldm1cossres2 36923 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅))
43elv 3451 . . . . . 6 (𝑥 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅)
54anbi1i 624 . . . . 5 ((𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
65exbii 1850 . . . 4 (∃𝑥(𝑥 ∈ dom ≀ (𝑅𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
72, 6bitri 274 . . 3 (∃𝑥 ∈ dom ≀ (𝑅𝐴)𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
81, 7bitrdi 286 . 2 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴))))
9 df-rex 3074 . . . 4 (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
109rexbii 3097 . . 3 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
11 rexcom4 3271 . . . 4 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
12 r19.41v 3185 . . . . 5 (∃𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ (∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1312exbii 1850 . . . 4 (∃𝑥𝑢𝐴 (𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1411, 13bitri 274 . . 3 (∃𝑢𝐴𝑥(𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
1510, 14bitri 274 . 2 (∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴) ↔ ∃𝑥(∃𝑢𝐴 𝑥 ∈ [𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
168, 15bitr4di 288 1 (𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3073  Vcvv 3445  dom cdm 5633  cres 5635  [cec 8646   / cqs 8647  ccoss 36634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ec 8650  df-qs 8654  df-coss 36873
This theorem is referenced by:  releldmqscoss  37122
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