Theorem eldmqs1cossres 35908

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 8348	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
2		df-rex 3144	. . . 4 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
3		eldm1cossres2 35716	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅))
4	3	elv 3499	. . . . . 6 ⊢ (𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅)
5	4	anbi1i 625	. . . . 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 277	. . 3 ⊢ (∃𝑥 ∈ dom ≀ (𝑅 ↾ 𝐴)𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
8	1, 7	syl6bb 289	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴))))
9		df-rex 3144	. . . 4 ⊢ (∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
10	9	rexbii 3247	. . 3 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
11		rexcom4 3249	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑥∃𝑢 ∈ 𝐴 (𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
12		r19.41v 3347	. . . . 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 277	. . 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 277	. 2 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥(∃𝑢 ∈ 𝐴 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))
16	8, 15	syl6bbr 291	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃wrex 3139  Vcvv 3494  dom cdm 5555   ↾ cres 5557  [cec 8287   / cqs 8288   ≀ ccoss 35468

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ec 8291  df-qs 8295  df-coss 35674

This theorem is referenced by:  releldmqscoss  35909