Theorem eldmqs1cossres 38676

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 8683	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
2		df-rex 3055	. . . 4 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
3		eldm1cossres2 38477	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅))
4	3	elv 3439	. . . . . 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 1849	. . . 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 3055	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
10	9	rexbii 3077	. . 3 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
11		rexcom4 3257	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
12		r19.41v 3160	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ (∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
13	12	exbii 1849	. . . 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 1541  ∃wex 1780   ∈ wcel 2110  ∃wrex 3054  Vcvv 3434  dom cdm 5614   ↾ cres 5616  [cec 8615   / cqs 8616   ≀ ccoss 38194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-11 2159  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8619  df-qs 8623  df-coss 38427

This theorem is referenced by:  releldmqscoss  38677