Theorem dmxrncnvepres2

Description: Domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 29-Jan-2026)

	Ref	Expression
Assertion	dmxrncnvepres2	⊢ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) = ( 𝐴 ∩ ( dom 𝑅 ∖ { ∅ } ) )

Proof

Step	Hyp	Ref	Expression
1		dmres	⊢ dom ( 𝑅 ↾ 𝐴 ) = ( 𝐴 ∩ dom 𝑅 )
2	1	difeq1i	⊢ ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } ) = ( ( 𝐴 ∩ dom 𝑅 ) ∖ { ∅ } )
3		dmxrncnvepres	⊢ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) = ( dom ( 𝑅 ↾ 𝐴 ) ∖ { ∅ } )
4		indif2	⊢ ( 𝐴 ∩ ( dom 𝑅 ∖ { ∅ } ) ) = ( ( 𝐴 ∩ dom 𝑅 ) ∖ { ∅ } )
5	2 3 4	3eqtr4i	⊢ dom ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) = ( 𝐴 ∩ ( dom 𝑅 ∖ { ∅ } ) )