Step |
Hyp |
Ref |
Expression |
1 |
|
dff13f.1 |
⊢ Ⅎ 𝑥 𝐹 |
2 |
|
dff13f.2 |
⊢ Ⅎ 𝑦 𝐹 |
3 |
|
dff13 |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) ) ) |
4 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑤 |
5 |
2 4
|
nffv |
⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑤 ) |
6 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑣 |
7 |
2 6
|
nffv |
⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑣 ) |
8 |
5 7
|
nfeq |
⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) |
9 |
|
nfv |
⊢ Ⅎ 𝑦 𝑤 = 𝑣 |
10 |
8 9
|
nfim |
⊢ Ⅎ 𝑦 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) |
11 |
|
nfv |
⊢ Ⅎ 𝑣 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) |
12 |
|
fveq2 |
⊢ ( 𝑣 = 𝑦 → ( 𝐹 ‘ 𝑣 ) = ( 𝐹 ‘ 𝑦 ) ) |
13 |
12
|
eqeq2d |
⊢ ( 𝑣 = 𝑦 → ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) ↔ ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) ) ) |
14 |
|
equequ2 |
⊢ ( 𝑣 = 𝑦 → ( 𝑤 = 𝑣 ↔ 𝑤 = 𝑦 ) ) |
15 |
13 14
|
imbi12d |
⊢ ( 𝑣 = 𝑦 → ( ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) ↔ ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) ) ) |
16 |
10 11 15
|
cbvralw |
⊢ ( ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) ↔ ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) ) |
17 |
16
|
ralbii |
⊢ ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) ↔ ∀ 𝑤 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) ) |
18 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
19 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑤 |
20 |
1 19
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑤 ) |
21 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑦 |
22 |
1 21
|
nffv |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 ) |
23 |
20 22
|
nfeq |
⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) |
24 |
|
nfv |
⊢ Ⅎ 𝑥 𝑤 = 𝑦 |
25 |
23 24
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) |
26 |
18 25
|
nfralw |
⊢ Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) |
27 |
|
nfv |
⊢ Ⅎ 𝑤 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) |
28 |
|
fveqeq2 |
⊢ ( 𝑤 = 𝑥 → ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) ) ) |
29 |
|
equequ1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 = 𝑦 ↔ 𝑥 = 𝑦 ) ) |
30 |
28 29
|
imbi12d |
⊢ ( 𝑤 = 𝑥 → ( ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
31 |
30
|
ralbidv |
⊢ ( 𝑤 = 𝑥 → ( ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
32 |
26 27 31
|
cbvralw |
⊢ ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑦 ) → 𝑤 = 𝑦 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
33 |
17 32
|
bitri |
⊢ ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
34 |
33
|
anbi2i |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑤 ∈ 𝐴 ∀ 𝑣 ∈ 𝐴 ( ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑣 ) → 𝑤 = 𝑣 ) ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
35 |
3 34
|
bitri |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |