Description: A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | rabfi | ⊢ ( 𝐴 ∈ Fin → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ Fin ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrab3 | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = ( 𝐴 ∩ { 𝑥 ∣ 𝜑 } ) | |
2 | infi | ⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∩ { 𝑥 ∣ 𝜑 } ) ∈ Fin ) | |
3 | 1 2 | eqeltrid | ⊢ ( 𝐴 ∈ Fin → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ Fin ) |