Theorem imassrn

Description: The image of a class is a subset of its range. Theorem 3.16(xi) of Monk1 p. 39. (Contributed by NM, 31-Mar-1995)

	Ref	Expression
Assertion	imassrn	⊢ ( 𝐴 “ 𝐵 ) ⊆ ran 𝐴

Proof

Step	Hyp	Ref	Expression
1		exsimpr	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) → ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 )
2	1	ss2abi	⊢ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) } ⊆ { 𝑦 ∣ ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 }
3		dfima3	⊢ ( 𝐴 “ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) }
4		dfrn3	⊢ ran 𝐴 = { 𝑦 ∣ ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 }
5	2 3 4	3sstr4i	⊢ ( 𝐴 “ 𝐵 ) ⊆ ran 𝐴