Step |
Hyp |
Ref |
Expression |
1 |
|
tposfn2 |
⊢ ( Rel 𝐴 → ( 𝐹 Fn 𝐴 → tpos 𝐹 Fn ◡ 𝐴 ) ) |
2 |
1
|
adantrd |
⊢ ( Rel 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → tpos 𝐹 Fn ◡ 𝐴 ) ) |
3 |
|
fndm |
⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 ) |
4 |
3
|
releqd |
⊢ ( 𝐹 Fn 𝐴 → ( Rel dom 𝐹 ↔ Rel 𝐴 ) ) |
5 |
4
|
biimparc |
⊢ ( ( Rel 𝐴 ∧ 𝐹 Fn 𝐴 ) → Rel dom 𝐹 ) |
6 |
|
rntpos |
⊢ ( Rel dom 𝐹 → ran tpos 𝐹 = ran 𝐹 ) |
7 |
5 6
|
syl |
⊢ ( ( Rel 𝐴 ∧ 𝐹 Fn 𝐴 ) → ran tpos 𝐹 = ran 𝐹 ) |
8 |
7
|
eqeq1d |
⊢ ( ( Rel 𝐴 ∧ 𝐹 Fn 𝐴 ) → ( ran tpos 𝐹 = 𝐵 ↔ ran 𝐹 = 𝐵 ) ) |
9 |
8
|
biimprd |
⊢ ( ( Rel 𝐴 ∧ 𝐹 Fn 𝐴 ) → ( ran 𝐹 = 𝐵 → ran tpos 𝐹 = 𝐵 ) ) |
10 |
9
|
expimpd |
⊢ ( Rel 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ran tpos 𝐹 = 𝐵 ) ) |
11 |
2 10
|
jcad |
⊢ ( Rel 𝐴 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) → ( tpos 𝐹 Fn ◡ 𝐴 ∧ ran tpos 𝐹 = 𝐵 ) ) ) |
12 |
|
df-fo |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ) ) |
13 |
|
df-fo |
⊢ ( tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ↔ ( tpos 𝐹 Fn ◡ 𝐴 ∧ ran tpos 𝐹 = 𝐵 ) ) |
14 |
11 12 13
|
3imtr4g |
⊢ ( Rel 𝐴 → ( 𝐹 : 𝐴 –onto→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –onto→ 𝐵 ) ) |