Description: A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | fndmeng | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → 𝐴 ≈ 𝐹 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnex | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → 𝐹 ∈ V ) | |
2 | fnfun | ⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 ) | |
3 | 2 | adantr | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → Fun 𝐹 ) |
4 | fundmeng | ⊢ ( ( 𝐹 ∈ V ∧ Fun 𝐹 ) → dom 𝐹 ≈ 𝐹 ) | |
5 | 1 3 4 | syl2anc | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → dom 𝐹 ≈ 𝐹 ) |
6 | fndm | ⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 ) | |
7 | 6 | breq1d | ⊢ ( 𝐹 Fn 𝐴 → ( dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹 ) ) |
8 | 7 | adantr | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → ( dom 𝐹 ≈ 𝐹 ↔ 𝐴 ≈ 𝐹 ) ) |
9 | 5 8 | mpbid | ⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶 ) → 𝐴 ≈ 𝐹 ) |