Metamath Proof Explorer
Description: The range of a function in maps-to notation. (Contributed by Glauco
Siliprandi, 17-Aug-2020)
|
|
Ref |
Expression |
|
Hypotheses |
elrnmptd.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
|
|
elrnmptd.x |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) |
|
|
elrnmptd.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
|
Assertion |
elrnmptd |
⊢ ( 𝜑 → 𝐶 ∈ ran 𝐹 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
elrnmptd.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
2 |
|
elrnmptd.x |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) |
3 |
|
elrnmptd.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑉 ) |
4 |
1
|
elrnmpt |
⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) ) |
5 |
3 4
|
syl |
⊢ ( 𝜑 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) ) |
6 |
2 5
|
mpbird |
⊢ ( 𝜑 → 𝐶 ∈ ran 𝐹 ) |