Theorem dmxrncnvepres

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

	Ref	Expression
Assertion	dmxrncnvepres	⊢ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) = ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } )

Proof

Step	Hyp	Ref	Expression
1		xrnres	⊢ ( ( 𝑅 ⋉ ◡ E ) ↾ 𝐴 ) = ( ( 𝑅 ↾ 𝐴 ) ⋉ ◡ E )
2		xrnres2	⊢ ( ( 𝑅 ⋉ ◡ E ) ↾ 𝐴 ) = ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) )
3	1 2	eqtr3i	⊢ ( ( 𝑅 ↾ 𝐴 ) ⋉ ◡ E ) = ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) )
4	3	dmeqi	⊢ dom ( ( 𝑅 ↾ 𝐴 ) ⋉ ◡ E ) = dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) )
5		dmxrncnvep	⊢ dom ( ( 𝑅 ↾ 𝐴 ) ⋉ ◡ E ) = ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } )
6	4 5	eqtr3i	⊢ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) = ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } )