Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → 𝐹 : 𝐴 –1-1→ 𝐵 ) |
2 |
|
relcnv |
⊢ Rel ◡ 𝐴 |
3 |
|
cnvf1o |
⊢ ( Rel ◡ 𝐴 → ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1-onto→ ◡ ◡ 𝐴 ) |
4 |
|
f1of1 |
⊢ ( ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1-onto→ ◡ ◡ 𝐴 → ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ ◡ ◡ 𝐴 ) |
5 |
2 3 4
|
mp2b |
⊢ ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ ◡ ◡ 𝐴 |
6 |
|
simpl |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → Rel 𝐴 ) |
7 |
|
dfrel2 |
⊢ ( Rel 𝐴 ↔ ◡ ◡ 𝐴 = 𝐴 ) |
8 |
6 7
|
sylib |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ◡ ◡ 𝐴 = 𝐴 ) |
9 |
|
f1eq3 |
⊢ ( ◡ ◡ 𝐴 = 𝐴 → ( ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ ◡ ◡ 𝐴 ↔ ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) ) |
10 |
8 9
|
syl |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ ◡ ◡ 𝐴 ↔ ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) ) |
11 |
5 10
|
mpbii |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) |
12 |
|
f1dm |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → dom 𝐹 = 𝐴 ) |
13 |
1 12
|
syl |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → dom 𝐹 = 𝐴 ) |
14 |
13
|
cnveqd |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ◡ dom 𝐹 = ◡ 𝐴 ) |
15 |
|
mpteq1 |
⊢ ( ◡ dom 𝐹 = ◡ 𝐴 → ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) = ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) ) |
16 |
|
f1eq1 |
⊢ ( ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) = ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) → ( ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ↔ ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) ) |
17 |
14 15 16
|
3syl |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ↔ ( 𝑥 ∈ ◡ 𝐴 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) ) |
18 |
11 17
|
mpbird |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) |
19 |
|
f1co |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) : ◡ 𝐴 –1-1→ 𝐴 ) → ( 𝐹 ∘ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) ) : ◡ 𝐴 –1-1→ 𝐵 ) |
20 |
1 18 19
|
syl2anc |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( 𝐹 ∘ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) ) : ◡ 𝐴 –1-1→ 𝐵 ) |
21 |
12
|
releqd |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ( Rel dom 𝐹 ↔ Rel 𝐴 ) ) |
22 |
21
|
biimparc |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → Rel dom 𝐹 ) |
23 |
|
dftpos2 |
⊢ ( Rel dom 𝐹 → tpos 𝐹 = ( 𝐹 ∘ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) ) ) |
24 |
|
f1eq1 |
⊢ ( tpos 𝐹 = ( 𝐹 ∘ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) ) → ( tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 ∘ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) ) : ◡ 𝐴 –1-1→ 𝐵 ) ) |
25 |
22 23 24
|
3syl |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → ( tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 ∘ ( 𝑥 ∈ ◡ dom 𝐹 ↦ ∪ ◡ { 𝑥 } ) ) : ◡ 𝐴 –1-1→ 𝐵 ) ) |
26 |
20 25
|
mpbird |
⊢ ( ( Rel 𝐴 ∧ 𝐹 : 𝐴 –1-1→ 𝐵 ) → tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ) |
27 |
26
|
ex |
⊢ ( Rel 𝐴 → ( 𝐹 : 𝐴 –1-1→ 𝐵 → tpos 𝐹 : ◡ 𝐴 –1-1→ 𝐵 ) ) |