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Metamath Proof Explorer

Theorem dfimafn

Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006)

Ref Expression
Assertion dfimafn ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( 𝐹 “ 𝐴 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 } )

Proof

Step Hyp Ref Expression
1 dfima2 ⊢ ( 𝐹 “ 𝐴 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑥 𝐹 𝑦 }
2 ssel ⊢ ( 𝐴 ⊆ dom 𝐹 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹 ) )
3 funbrfvb ⊢ ( ( Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ 𝑥 𝐹 𝑦 ) )
4 3 ex ⊢ ( Fun 𝐹 → ( 𝑥 ∈ dom 𝐹 → ( ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ 𝑥 𝐹 𝑦 ) ) )
5 2 4 syl9r ⊢ ( Fun 𝐹 → ( 𝐴 ⊆ dom 𝐹 → ( 𝑥 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ 𝑥 𝐹 𝑦 ) ) ) )
6 5 imp31 ⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ 𝑥 𝐹 𝑦 ) )
7 6 rexbidva ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ ∃ 𝑥 ∈ 𝐴 𝑥 𝐹 𝑦 ) )
8 7 abbidv ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑥 𝐹 𝑦 } )
9 1 8 eqtr4id ⊢ ( ( Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹 ) → ( 𝐹 “ 𝐴 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 } )