Theorem eldmqs1cossres 38644

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	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))

Distinct variable groups:   𝑢,𝐴,𝑥   𝑢,𝐵,𝑥   𝑢,𝑅,𝑥

Allowed substitution hints:   𝑉(𝑥,𝑢)

Proof of Theorem eldmqs1cossres

Step	Hyp	Ref	Expression
1		elqsg 8714	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
2		df-rex 3054	. . . 4 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
3		eldm1cossres2 38445	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅))
4	3	elv 3449	. . . . . 6 ⊢ (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅)
5	4	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
6	5	exbii 1848	. . . 4 ⊢ (∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
7	2, 6	bitri 275	. . 3 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
8	1, 7	bitrdi 287	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴))))
9		df-rex 3054	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
10	9	rexbii 3076	. . 3 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
11		rexcom4 3262	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
12		r19.41v 3165	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
13	12	exbii 1848	. . . 4 ⊢ (∃𝑥∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
14	11, 13	bitri 275	. . 3 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
15	10, 14	bitri 275	. 2 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
16	8, 15	bitr4di 289	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  Vcvv 3444  dom cdm 5631   ↾ cres 5633  [cec 8646   / cqs 8647   ≀ ccoss 38162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654  df-coss 38395

This theorem is referenced by:  releldmqscoss  38645