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Metamath Proof Explorer

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 ) )