Step |
Hyp |
Ref |
Expression |
1 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) |
2 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐹 “ 𝑋 ) |
3 |
|
nfrab1 |
⊢ Ⅎ 𝑦 { 𝑦 ∈ 𝐵 ∣ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 } |
4 |
|
ffn |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 ) |
5 |
|
fvelimab |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝑋 ) ↔ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
6 |
5
|
anbi2d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ) ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) ) |
7 |
4 6
|
sylan |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ) ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) ) |
8 |
|
fimass |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 “ 𝑋 ) ⊆ 𝐵 ) |
9 |
8
|
adantr |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝐹 “ 𝑋 ) ⊆ 𝐵 ) |
10 |
9
|
sseld |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝑋 ) → 𝑦 ∈ 𝐵 ) ) |
11 |
10
|
pm4.71rd |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝑋 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ( 𝐹 “ 𝑋 ) ) ) ) |
12 |
|
rabid |
⊢ ( 𝑦 ∈ { 𝑦 ∈ 𝐵 ∣ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 } ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
13 |
12
|
a1i |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑦 ∈ { 𝑦 ∈ 𝐵 ∣ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 } ↔ ( 𝑦 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) ) |
14 |
7 11 13
|
3bitr4d |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝑋 ) ↔ 𝑦 ∈ { 𝑦 ∈ 𝐵 ∣ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 } ) ) |
15 |
1 2 3 14
|
eqrd |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝐹 “ 𝑋 ) = { 𝑦 ∈ 𝐵 ∣ ∃ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = 𝑦 } ) |