Theorem eldmqs1cossres 39644

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 8768	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
2		df-rex 3088	. . . 4 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
3		eldm1cossres2 39451	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅))
4	3	elv 3456	. . . . . 6 ⊢ (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅)
5	4	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
6	5	exbii 1881	. . . 4 ⊢ (∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
7	2, 6	bitri 278	. . 3 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
8	1, 7	bitrdi 290	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴))))
9		df-rex 3088	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
10	9	rexbii 3110	. . 3 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
11		rexcom4 3290	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
12		r19.41v 3193	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
13	12	exbii 1881	. . . 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 278	. . 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 278	. 2 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
16	8, 15	bitr4di 292	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  dom cdm 5651   ↾ cres 5653  [cec 8699   / cqs 8700   ≀ ccoss 39083

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8703  df-qs 8707  df-coss 39401

This theorem is used by:  releldmqscoss  39645