Step |
Hyp |
Ref |
Expression |
1 |
|
f1f |
⊢ ( 𝐹 : 𝐴 –1-1→ { 𝐵 } → 𝐹 : 𝐴 ⟶ { 𝐵 } ) |
2 |
|
fvconst |
⊢ ( ( 𝐹 : 𝐴 ⟶ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = 𝐵 ) |
3 |
2
|
3adant3 |
⊢ ( ( 𝐹 : 𝐴 ⟶ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = 𝐵 ) |
4 |
|
fvconst |
⊢ ( ( 𝐹 : 𝐴 ⟶ { 𝐵 } ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) = 𝐵 ) |
5 |
4
|
3adant2 |
⊢ ( ( 𝐹 : 𝐴 ⟶ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) = 𝐵 ) |
6 |
3 5
|
eqtr4d |
⊢ ( ( 𝐹 : 𝐴 ⟶ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) |
7 |
1 6
|
syl3an1 |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) |
8 |
|
f1veqaeq |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
9 |
8
|
3impb |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) → 𝑦 = 𝑧 ) ) |
10 |
7 9
|
mpd |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ) → 𝑦 = 𝑧 ) |
11 |
10
|
3expia |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ) → ( 𝑧 ∈ 𝐴 → 𝑦 = 𝑧 ) ) |
12 |
11
|
ralrimiv |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝑦 ∈ 𝐴 ) → ∀ 𝑧 ∈ 𝐴 𝑦 = 𝑧 ) |
13 |
12
|
reximdva0 |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝐴 ≠ ∅ ) → ∃ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 𝑦 = 𝑧 ) |
14 |
|
issn |
⊢ ( ∃ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 𝑦 = 𝑧 → ∃ 𝑥 𝐴 = { 𝑥 } ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐹 : 𝐴 –1-1→ { 𝐵 } ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 𝐴 = { 𝑥 } ) |