Theorem releldmqscoss

Description: Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021)

	Ref	Expression
Assertion	releldmqscoss	⊢ ( 𝐴 ∈ 𝑉 → ( Rel 𝑅 → ( 𝐴 ∈ ( dom ≀ 𝑅 / ≀ 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eldmqs1cossres	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ( dom ≀ ( 𝑅 ↾ dom 𝑅 ) / ≀ ( 𝑅 ↾ dom 𝑅 ) ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ ( 𝑅 ↾ dom 𝑅 ) ) )
2	1	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( 𝐴 ∈ ( dom ≀ ( 𝑅 ↾ dom 𝑅 ) / ≀ ( 𝑅 ↾ dom 𝑅 ) ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ ( 𝑅 ↾ dom 𝑅 ) ) )
3		resdm	⊢ ( Rel 𝑅 → ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )
4	3	cosseqd	⊢ ( Rel 𝑅 → ≀ ( 𝑅 ↾ dom 𝑅 ) = ≀ 𝑅 )
5	4	dmqseqd	⊢ ( Rel 𝑅 → ( dom ≀ ( 𝑅 ↾ dom 𝑅 ) / ≀ ( 𝑅 ↾ dom 𝑅 ) ) = ( dom ≀ 𝑅 / ≀ 𝑅 ) )
6	5	eleq2d	⊢ ( Rel 𝑅 → ( 𝐴 ∈ ( dom ≀ ( 𝑅 ↾ dom 𝑅 ) / ≀ ( 𝑅 ↾ dom 𝑅 ) ) ↔ 𝐴 ∈ ( dom ≀ 𝑅 / ≀ 𝑅 ) ) )
7	6	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( 𝐴 ∈ ( dom ≀ ( 𝑅 ↾ dom 𝑅 ) / ≀ ( 𝑅 ↾ dom 𝑅 ) ) ↔ 𝐴 ∈ ( dom ≀ 𝑅 / ≀ 𝑅 ) ) )
8	4	eceq2d	⊢ ( Rel 𝑅 → [ 𝑥 ] ≀ ( 𝑅 ↾ dom 𝑅 ) = [ 𝑥 ] ≀ 𝑅 )
9	8	eqeq2d	⊢ ( Rel 𝑅 → ( 𝐴 = [ 𝑥 ] ≀ ( 𝑅 ↾ dom 𝑅 ) ↔ 𝐴 = [ 𝑥 ] ≀ 𝑅 ) )
10	9	2rexbidv	⊢ ( Rel 𝑅 → ( ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ ( 𝑅 ↾ dom 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ 𝑅 ) )
11	10	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ ( 𝑅 ↾ dom 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ 𝑅 ) )
12	2 7 11	3bitr3d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( 𝐴 ∈ ( dom ≀ 𝑅 / ≀ 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ 𝑅 ) )
13	12	ex	⊢ ( 𝐴 ∈ 𝑉 → ( Rel 𝑅 → ( 𝐴 ∈ ( dom ≀ 𝑅 / ≀ 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑥 ] ≀ 𝑅 ) ) )