Theorem releldmqs

Description: Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021)

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

Proof

Step	Hyp	Ref	Expression
1		resdm	⊢ ( Rel 𝑅 → ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )
2	1	dmqseqd	⊢ ( Rel 𝑅 → ( dom ( 𝑅 ↾ dom 𝑅 ) / ( 𝑅 ↾ dom 𝑅 ) ) = ( dom 𝑅 / 𝑅 ) )
3	2	eleq2d	⊢ ( Rel 𝑅 → ( 𝐴 ∈ ( dom ( 𝑅 ↾ dom 𝑅 ) / ( 𝑅 ↾ dom 𝑅 ) ) ↔ 𝐴 ∈ ( dom 𝑅 / 𝑅 ) ) )
4	3	adantl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( 𝐴 ∈ ( dom ( 𝑅 ↾ dom 𝑅 ) / ( 𝑅 ↾ dom 𝑅 ) ) ↔ 𝐴 ∈ ( dom 𝑅 / 𝑅 ) ) )
5		eldmqsres2	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ( dom ( 𝑅 ↾ dom 𝑅 ) / ( 𝑅 ↾ dom 𝑅 ) ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑢 ] 𝑅 ) )
6	5	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( 𝐴 ∈ ( dom ( 𝑅 ↾ dom 𝑅 ) / ( 𝑅 ↾ dom 𝑅 ) ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑢 ] 𝑅 ) )
7	4 6	bitr3d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Rel 𝑅 ) → ( 𝐴 ∈ ( dom 𝑅 / 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑢 ] 𝑅 ) )
8	7	ex	⊢ ( 𝐴 ∈ 𝑉 → ( Rel 𝑅 → ( 𝐴 ∈ ( dom 𝑅 / 𝑅 ) ↔ ∃ 𝑢 ∈ dom 𝑅 ∃ 𝑥 ∈ [ 𝑢 ] 𝑅 𝐴 = [ 𝑢 ] 𝑅 ) ) )