Theorem eldmxrncnvepres

Description: Element of the domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 23-Nov-2025)

	Ref	Expression
Assertion	eldmxrncnvepres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ∧ [ 𝐵 ] 𝑅 ≠ ∅ ) ) )

Proof

Step	Hyp	Ref	Expression
1		eldmres3	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ( 𝑅 ↾ 𝐴 ) ↔ ( 𝐵 ∈ 𝐴 ∧ [ 𝐵 ] 𝑅 ≠ ∅ ) ) )
2	1	anbi1d	⊢ ( 𝐵 ∈ 𝑉 → ( ( 𝐵 ∈ dom ( 𝑅 ↾ 𝐴 ) ∧ 𝐵 ≠ ∅ ) ↔ ( ( 𝐵 ∈ 𝐴 ∧ [ 𝐵 ] 𝑅 ≠ ∅ ) ∧ 𝐵 ≠ ∅ ) ) )
3		dmxrncnvepres	⊢ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) = ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } )
4	3	eleq2i	⊢ ( 𝐵 ∈ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ↔ 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } ) )
5		eldifsn	⊢ ( 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } ) ↔ ( 𝐵 ∈ dom ( 𝑅 ↾ 𝐴 ) ∧ 𝐵 ≠ ∅ ) )
6	4 5	bitri	⊢ ( 𝐵 ∈ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ↔ ( 𝐵 ∈ dom ( 𝑅 ↾ 𝐴 ) ∧ 𝐵 ≠ ∅ ) )
7		3anan32	⊢ ( ( 𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ∧ [ 𝐵 ] 𝑅 ≠ ∅ ) ↔ ( ( 𝐵 ∈ 𝐴 ∧ [ 𝐵 ] 𝑅 ≠ ∅ ) ∧ 𝐵 ≠ ∅ ) )
8	2 6 7	3bitr4g	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅ ∧ [ 𝐵 ] 𝑅 ≠ ∅ ) ) )