Step |
Hyp |
Ref |
Expression |
1 |
|
f1f |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
|
fo2ndf |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 2nd ↾ 𝐹 ) : 𝐹 –onto→ ran 𝐹 ) |
3 |
1 2
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 2nd ↾ 𝐹 ) : 𝐹 –onto→ ran 𝐹 ) |
4 |
|
f2ndf |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 2nd ↾ 𝐹 ) : 𝐹 ⟶ 𝐵 ) |
5 |
1 4
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 2nd ↾ 𝐹 ) : 𝐹 ⟶ 𝐵 ) |
6 |
|
fssxp |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) |
7 |
1 6
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 ⊆ ( 𝐴 × 𝐵 ) ) |
8 |
|
ssel2 |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ 𝑥 ∈ 𝐹 ) → 𝑥 ∈ ( 𝐴 × 𝐵 ) ) |
9 |
|
elxp2 |
⊢ ( 𝑥 ∈ ( 𝐴 × 𝐵 ) ↔ ∃ 𝑎 ∈ 𝐴 ∃ 𝑣 ∈ 𝐵 𝑥 = 〈 𝑎 , 𝑣 〉 ) |
10 |
8 9
|
sylib |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ 𝑥 ∈ 𝐹 ) → ∃ 𝑎 ∈ 𝐴 ∃ 𝑣 ∈ 𝐵 𝑥 = 〈 𝑎 , 𝑣 〉 ) |
11 |
|
ssel2 |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ∈ ( 𝐴 × 𝐵 ) ) |
12 |
|
elxp2 |
⊢ ( 𝑦 ∈ ( 𝐴 × 𝐵 ) ↔ ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 ) |
13 |
11 12
|
sylib |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ 𝑦 ∈ 𝐹 ) → ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 ) |
14 |
10 13
|
anim12dan |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑣 ∈ 𝐵 𝑥 = 〈 𝑎 , 𝑣 〉 ∧ ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 ) ) |
15 |
|
fvres |
⊢ ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 → ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( 2nd ‘ 〈 𝑎 , 𝑣 〉 ) ) |
16 |
15
|
ad2antrr |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( 2nd ‘ 〈 𝑎 , 𝑣 〉 ) ) |
17 |
|
fvres |
⊢ ( 〈 𝑏 , 𝑤 〉 ∈ 𝐹 → ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) = ( 2nd ‘ 〈 𝑏 , 𝑤 〉 ) ) |
18 |
17
|
ad2antlr |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) = ( 2nd ‘ 〈 𝑏 , 𝑤 〉 ) ) |
19 |
16 18
|
eqeq12d |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) ↔ ( 2nd ‘ 〈 𝑎 , 𝑣 〉 ) = ( 2nd ‘ 〈 𝑏 , 𝑤 〉 ) ) ) |
20 |
|
vex |
⊢ 𝑎 ∈ V |
21 |
|
vex |
⊢ 𝑣 ∈ V |
22 |
20 21
|
op2nd |
⊢ ( 2nd ‘ 〈 𝑎 , 𝑣 〉 ) = 𝑣 |
23 |
|
vex |
⊢ 𝑏 ∈ V |
24 |
|
vex |
⊢ 𝑤 ∈ V |
25 |
23 24
|
op2nd |
⊢ ( 2nd ‘ 〈 𝑏 , 𝑤 〉 ) = 𝑤 |
26 |
22 25
|
eqeq12i |
⊢ ( ( 2nd ‘ 〈 𝑎 , 𝑣 〉 ) = ( 2nd ‘ 〈 𝑏 , 𝑤 〉 ) ↔ 𝑣 = 𝑤 ) |
27 |
|
f1fun |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → Fun 𝐹 ) |
28 |
|
funopfv |
⊢ ( Fun 𝐹 → ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 → ( 𝐹 ‘ 𝑎 ) = 𝑣 ) ) |
29 |
|
funopfv |
⊢ ( Fun 𝐹 → ( 〈 𝑏 , 𝑤 〉 ∈ 𝐹 → ( 𝐹 ‘ 𝑏 ) = 𝑤 ) ) |
30 |
28 29
|
anim12d |
⊢ ( Fun 𝐹 → ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) → ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) ) ) |
31 |
27 30
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) → ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) ) ) |
32 |
|
eqcom |
⊢ ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ↔ 𝑣 = ( 𝐹 ‘ 𝑎 ) ) |
33 |
32
|
biimpi |
⊢ ( ( 𝐹 ‘ 𝑎 ) = 𝑣 → 𝑣 = ( 𝐹 ‘ 𝑎 ) ) |
34 |
|
eqcom |
⊢ ( ( 𝐹 ‘ 𝑏 ) = 𝑤 ↔ 𝑤 = ( 𝐹 ‘ 𝑏 ) ) |
35 |
34
|
biimpi |
⊢ ( ( 𝐹 ‘ 𝑏 ) = 𝑤 → 𝑤 = ( 𝐹 ‘ 𝑏 ) ) |
36 |
33 35
|
eqeqan12d |
⊢ ( ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) → ( 𝑣 = 𝑤 ↔ ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) ) ) |
37 |
|
simpl |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) → 𝑎 ∈ 𝐴 ) |
38 |
|
simpl |
⊢ ( ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) → 𝑏 ∈ 𝐴 ) |
39 |
37 38
|
anim12i |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ) ) |
40 |
|
f1veqaeq |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) |
41 |
39 40
|
sylan2 |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) |
42 |
|
opeq12 |
⊢ ( ( 𝑎 = 𝑏 ∧ 𝑣 = 𝑤 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) |
43 |
42
|
ex |
⊢ ( 𝑎 = 𝑏 → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) |
44 |
41 43
|
syl6 |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
45 |
44
|
com23 |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( 𝑣 = 𝑤 → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
46 |
45
|
ex |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑣 = 𝑤 → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
47 |
46
|
com14 |
⊢ ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑏 ) → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑣 = 𝑤 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
48 |
36 47
|
syl6bi |
⊢ ( ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) → ( 𝑣 = 𝑤 → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑣 = 𝑤 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) ) |
49 |
48
|
com14 |
⊢ ( 𝑣 = 𝑤 → ( 𝑣 = 𝑤 → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) ) |
50 |
49
|
pm2.43i |
⊢ ( 𝑣 = 𝑤 → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
51 |
50
|
com14 |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
52 |
51
|
com23 |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 𝐹 ‘ 𝑎 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑏 ) = 𝑤 ) → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
53 |
31 52
|
syld |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
54 |
53
|
com13 |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
55 |
54
|
impcom |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 𝑣 = 𝑤 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
56 |
55
|
com23 |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( 𝑣 = 𝑤 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
57 |
26 56
|
syl5bi |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( 2nd ‘ 〈 𝑎 , 𝑣 〉 ) = ( 2nd ‘ 〈 𝑏 , 𝑤 〉 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
58 |
19 57
|
sylbid |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
59 |
58
|
com23 |
⊢ ( ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ∧ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
60 |
59
|
ex |
⊢ ( ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
61 |
60
|
adantl |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ) → ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
62 |
61
|
com12 |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
63 |
62
|
ad4ant13 |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
64 |
|
eleq1 |
⊢ ( 𝑥 = 〈 𝑎 , 𝑣 〉 → ( 𝑥 ∈ 𝐹 ↔ 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ) ) |
65 |
64
|
ad2antlr |
⊢ ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑥 ∈ 𝐹 ↔ 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ) ) |
66 |
|
eleq1 |
⊢ ( 𝑦 = 〈 𝑏 , 𝑤 〉 → ( 𝑦 ∈ 𝐹 ↔ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ) |
67 |
65 66
|
bi2anan9 |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ↔ ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ) ) |
68 |
67
|
anbi2d |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) ↔ ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 〈 𝑎 , 𝑣 〉 ∈ 𝐹 ∧ 〈 𝑏 , 𝑤 〉 ∈ 𝐹 ) ) ) ) |
69 |
|
fveq2 |
⊢ ( 𝑥 = 〈 𝑎 , 𝑣 〉 → ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) ) |
70 |
69
|
ad2antlr |
⊢ ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) ) |
71 |
|
fveq2 |
⊢ ( 𝑦 = 〈 𝑏 , 𝑤 〉 → ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) ) |
72 |
70 71
|
eqeqan12d |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) ↔ ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) ) ) |
73 |
|
simpllr |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → 𝑥 = 〈 𝑎 , 𝑣 〉 ) |
74 |
|
simpr |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → 𝑦 = 〈 𝑏 , 𝑤 〉 ) |
75 |
73 74
|
eqeq12d |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( 𝑥 = 𝑦 ↔ 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) |
76 |
72 75
|
imbi12d |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ↔ ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) |
77 |
76
|
imbi2d |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ↔ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑎 , 𝑣 〉 ) = ( ( 2nd ↾ 𝐹 ) ‘ 〈 𝑏 , 𝑤 〉 ) → 〈 𝑎 , 𝑣 〉 = 〈 𝑏 , 𝑤 〉 ) ) ) ) |
78 |
63 68 77
|
3imtr4d |
⊢ ( ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) ∧ 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) |
79 |
78
|
ex |
⊢ ( ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) ∧ ( 𝑏 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) → ( 𝑦 = 〈 𝑏 , 𝑤 〉 → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) ) |
80 |
79
|
rexlimdvva |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) ∧ 𝑥 = 〈 𝑎 , 𝑣 〉 ) → ( ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) ) |
81 |
80
|
ex |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵 ) → ( 𝑥 = 〈 𝑎 , 𝑣 〉 → ( ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) ) ) |
82 |
81
|
rexlimivv |
⊢ ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑣 ∈ 𝐵 𝑥 = 〈 𝑎 , 𝑣 〉 → ( ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) ) |
83 |
82
|
imp |
⊢ ( ( ∃ 𝑎 ∈ 𝐴 ∃ 𝑣 ∈ 𝐵 𝑥 = 〈 𝑎 , 𝑣 〉 ∧ ∃ 𝑏 ∈ 𝐴 ∃ 𝑤 ∈ 𝐵 𝑦 = 〈 𝑏 , 𝑤 〉 ) → ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) |
84 |
14 83
|
mpcom |
⊢ ( ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) ∧ ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
85 |
84
|
ex |
⊢ ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) → ( ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) |
86 |
85
|
com23 |
⊢ ( 𝐹 ⊆ ( 𝐴 × 𝐵 ) → ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) ) |
87 |
7 86
|
mpcom |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( ( 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹 ) → ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
88 |
87
|
ralrimivv |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ∀ 𝑥 ∈ 𝐹 ∀ 𝑦 ∈ 𝐹 ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
89 |
|
dff13 |
⊢ ( ( 2nd ↾ 𝐹 ) : 𝐹 –1-1→ 𝐵 ↔ ( ( 2nd ↾ 𝐹 ) : 𝐹 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐹 ∀ 𝑦 ∈ 𝐹 ( ( ( 2nd ↾ 𝐹 ) ‘ 𝑥 ) = ( ( 2nd ↾ 𝐹 ) ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
90 |
5 88 89
|
sylanbrc |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 2nd ↾ 𝐹 ) : 𝐹 –1-1→ 𝐵 ) |
91 |
|
df-f1 |
⊢ ( ( 2nd ↾ 𝐹 ) : 𝐹 –1-1→ 𝐵 ↔ ( ( 2nd ↾ 𝐹 ) : 𝐹 ⟶ 𝐵 ∧ Fun ◡ ( 2nd ↾ 𝐹 ) ) ) |
92 |
91
|
simprbi |
⊢ ( ( 2nd ↾ 𝐹 ) : 𝐹 –1-1→ 𝐵 → Fun ◡ ( 2nd ↾ 𝐹 ) ) |
93 |
90 92
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → Fun ◡ ( 2nd ↾ 𝐹 ) ) |
94 |
|
dff1o3 |
⊢ ( ( 2nd ↾ 𝐹 ) : 𝐹 –1-1-onto→ ran 𝐹 ↔ ( ( 2nd ↾ 𝐹 ) : 𝐹 –onto→ ran 𝐹 ∧ Fun ◡ ( 2nd ↾ 𝐹 ) ) ) |
95 |
3 93 94
|
sylanbrc |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( 2nd ↾ 𝐹 ) : 𝐹 –1-1-onto→ ran 𝐹 ) |