Theorem dfima2

Description: Alternate definition of image. Compare definition (d) of Enderton p. 44. (Contributed by NM, 19-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Proof shortened by Peter Mazsa, 2-Oct-2022)

	Ref	Expression
Assertion	dfima2	⊢ ( 𝐴 “ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑥 𝐴 𝑦 }

Proof

Step	Hyp	Ref	Expression
1		df-ima	⊢ ( 𝐴 “ 𝐵 ) = ran ( 𝐴 ↾ 𝐵 )
2		dfrn2	⊢ ran ( 𝐴 ↾ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 𝑥 ( 𝐴 ↾ 𝐵 ) 𝑦 }
3		brres	⊢ ( 𝑦 ∈ V → ( 𝑥 ( 𝐴 ↾ 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 𝐴 𝑦 ) ) )
4	3	elv	⊢ ( 𝑥 ( 𝐴 ↾ 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 𝐴 𝑦 ) )
5	4	exbii	⊢ ( ∃ 𝑥 𝑥 ( 𝐴 ↾ 𝐵 ) 𝑦 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 𝐴 𝑦 ) )
6		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝑥 𝐴 𝑦 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑥 𝐴 𝑦 ) )
7	5 6	bitr4i	⊢ ( ∃ 𝑥 𝑥 ( 𝐴 ↾ 𝐵 ) 𝑦 ↔ ∃ 𝑥 ∈ 𝐵 𝑥 𝐴 𝑦 )
8	7	abbii	⊢ { 𝑦 ∣ ∃ 𝑥 𝑥 ( 𝐴 ↾ 𝐵 ) 𝑦 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑥 𝐴 𝑦 }
9	1 2 8	3eqtri	⊢ ( 𝐴 “ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑥 𝐴 𝑦 }