Theorem eldmqs1cossres 39118

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 8707	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
2		df-rex 3065	. . . 4 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
3		eldm1cossres2 38925	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅))
4	3	elv 3437	. . . . . 6 ⊢ (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅)
5	4	anbi1i 630	. . . . 5 ⊢ ((𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
6	5	exbii 1855	. . . 4 ⊢ (∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
7	2, 6	bitri 276	. . 3 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
8	1, 7	bitrdi 288	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴))))
9		df-rex 3065	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
10	9	rexbii 3087	. . 3 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
11		rexcom4 3267	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
12		r19.41v 3170	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
13	12	exbii 1855	. . . 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 276	. . 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 276	. 2 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
16	8, 15	bitr4di 290	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃wrex 3064  Vcvv 3432  dom cdm 5625   ↾ cres 5627  [cec 8638   / cqs 8639   ≀ ccoss 38557

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8642  df-qs 8646  df-coss 38875

This theorem is referenced by:  releldmqscoss  39119