Theorem imaindm

Description: The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021)

	Ref	Expression
Assertion	imaindm	⊢ ( 𝑅 “ 𝐴 ) = ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑦 ∈ V
2		vex	⊢ 𝑥 ∈ V
3	1 2	breldm	⊢ ( 𝑦 𝑅 𝑥 → 𝑦 ∈ dom 𝑅 )
4	3	pm4.71ri	⊢ ( 𝑦 𝑅 𝑥 ↔ ( 𝑦 ∈ dom 𝑅 ∧ 𝑦 𝑅 𝑥 ) )
5	4	rexbii	⊢ ( ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑦 ∈ dom 𝑅 ∧ 𝑦 𝑅 𝑥 ) )
6		rexin	⊢ ( ∃ 𝑦 ∈ ( 𝐴 ∩ dom 𝑅 ) 𝑦 𝑅 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑦 ∈ dom 𝑅 ∧ 𝑦 𝑅 𝑥 ) )
7	5 6	bitr4i	⊢ ( ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ↔ ∃ 𝑦 ∈ ( 𝐴 ∩ dom 𝑅 ) 𝑦 𝑅 𝑥 )
8	2	elima	⊢ ( 𝑥 ∈ ( 𝑅 “ 𝐴 ) ↔ ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 )
9	2	elima	⊢ ( 𝑥 ∈ ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) ) ↔ ∃ 𝑦 ∈ ( 𝐴 ∩ dom 𝑅 ) 𝑦 𝑅 𝑥 )
10	7 8 9	3bitr4i	⊢ ( 𝑥 ∈ ( 𝑅 “ 𝐴 ) ↔ 𝑥 ∈ ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) ) )
11	10	eqriv	⊢ ( 𝑅 “ 𝐴 ) = ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) )