Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 31-Aug-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | elfunsALTV2 | ⊢ ( 𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfunsALTV | ⊢ ( 𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ) | |
2 | cosselcnvrefrels2 | ⊢ ( ≀ 𝐹 ∈ CnvRefRels ↔ ( ≀ 𝐹 ⊆ I ∧ ≀ 𝐹 ∈ Rels ) ) | |
3 | cosselrels | ⊢ ( 𝐹 ∈ Rels → ≀ 𝐹 ∈ Rels ) | |
4 | 3 | biantrud | ⊢ ( 𝐹 ∈ Rels → ( ≀ 𝐹 ⊆ I ↔ ( ≀ 𝐹 ⊆ I ∧ ≀ 𝐹 ∈ Rels ) ) ) |
5 | 2 4 | bitr4id | ⊢ ( 𝐹 ∈ Rels → ( ≀ 𝐹 ∈ CnvRefRels ↔ ≀ 𝐹 ⊆ I ) ) |
6 | 5 | pm5.32ri | ⊢ ( ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ) ) |
7 | 1 6 | bitri | ⊢ ( 𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ⊆ I ∧ 𝐹 ∈ Rels ) ) |