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Metamath Proof Explorer

Theorem ffnfv

Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003)

Ref Expression
Assertion ffnfv ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ffn ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
2 ffvelcdm ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
3 2 ralrimiva ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
4 1 3 jca ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
5 simpl ⊢ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → 𝐹 Fn 𝐴 )
6 fvelrnb ⊢ ( 𝐹 Fn 𝐴 → ( 𝑦 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
7 6 biimpd ⊢ ( 𝐹 Fn 𝐴 → ( 𝑦 ∈ ran 𝐹 → ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) )
8 nfra1 ⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵
9 nfv ⊢ Ⅎ 𝑥 𝑦 ∈ 𝐵
10 rsp ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
11 eleq1 ⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
12 11 biimpcd ⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 → ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → 𝑦 ∈ 𝐵 ) )
13 10 12 syl6 ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → 𝑦 ∈ 𝐵 ) ) )
14 8 9 13 rexlimd ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 → 𝑦 ∈ 𝐵 ) )
15 7 14 sylan9 ⊢ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → ( 𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐵 ) )
16 15 ssrdv ⊢ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → ran 𝐹 ⊆ 𝐵 )
17 df-f ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
18 5 16 17 sylanbrc ⊢ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 )
19 4 18 impbii ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )