Description: Subclass theorem for function. (Contributed by NM, 16-Aug-1994) (Proof shortened by Mario Carneiro, 24-Jun-2014) (Revised by Peter Mazsa, 22-Sep-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | funALTVss | ⊢ ( 𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossss | ⊢ ( 𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵 ) | |
2 | sstr2 | ⊢ ( ≀ 𝐴 ⊆ ≀ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ) ) | |
3 | 1 2 | syl | ⊢ ( 𝐴 ⊆ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ) ) |
4 | relss | ⊢ ( 𝐴 ⊆ 𝐵 → ( Rel 𝐵 → Rel 𝐴 ) ) | |
5 | 3 4 | anim12d | ⊢ ( 𝐴 ⊆ 𝐵 → ( ( ≀ 𝐵 ⊆ I ∧ Rel 𝐵 ) → ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴 ) ) ) |
6 | dffunALTV2 | ⊢ ( FunALTV 𝐵 ↔ ( ≀ 𝐵 ⊆ I ∧ Rel 𝐵 ) ) | |
7 | dffunALTV2 | ⊢ ( FunALTV 𝐴 ↔ ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴 ) ) | |
8 | 5 6 7 | 3imtr4g | ⊢ ( 𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴 ) ) |