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Metamath Proof Explorer

Theorem elrnmpt1d

Description: Elementhood in an image set. Deducion version of elrnmpt1 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses elrnmpt1d.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
elrnmpt1d.2 ⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
elrnmpt1d.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
Assertion elrnmpt1d ( 𝜑 → 𝐵 ∈ ran 𝐹 )

Proof

Step Hyp Ref Expression
1 elrnmpt1d.1 ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2 elrnmpt1d.2 ⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
3 elrnmpt1d.3 ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
4 1 elrnmpt1 ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ ran 𝐹 )
5 2 3 4 syl2anc ⊢ ( 𝜑 → 𝐵 ∈ ran 𝐹 )