Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrrn.t |
⊢ 𝑇 = ( pmTrsp ‘ 𝐷 ) |
2 |
|
pmtrrn.r |
⊢ 𝑅 = ran 𝑇 |
3 |
|
eqid |
⊢ dom ( 𝐹 ∖ I ) = dom ( 𝐹 ∖ I ) |
4 |
1 2 3
|
pmtrfrn |
⊢ ( 𝐹 ∈ 𝑅 → ( ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) ) |
5 |
|
simpl1 |
⊢ ( ( ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) → 𝐷 ∈ V ) |
6 |
4 5
|
syl |
⊢ ( 𝐹 ∈ 𝑅 → 𝐷 ∈ V ) |
7 |
1 2
|
pmtrff1o |
⊢ ( 𝐹 ∈ 𝑅 → 𝐹 : 𝐷 –1-1-onto→ 𝐷 ) |
8 |
|
simpl3 |
⊢ ( ( ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) → dom ( 𝐹 ∖ I ) ≈ 2o ) |
9 |
4 8
|
syl |
⊢ ( 𝐹 ∈ 𝑅 → dom ( 𝐹 ∖ I ) ≈ 2o ) |
10 |
6 7 9
|
3jca |
⊢ ( 𝐹 ∈ 𝑅 → ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ) |
11 |
|
simp2 |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → 𝐹 : 𝐷 –1-1-onto→ 𝐷 ) |
12 |
|
difss |
⊢ ( 𝐹 ∖ I ) ⊆ 𝐹 |
13 |
|
dmss |
⊢ ( ( 𝐹 ∖ I ) ⊆ 𝐹 → dom ( 𝐹 ∖ I ) ⊆ dom 𝐹 ) |
14 |
12 13
|
ax-mp |
⊢ dom ( 𝐹 ∖ I ) ⊆ dom 𝐹 |
15 |
|
f1odm |
⊢ ( 𝐹 : 𝐷 –1-1-onto→ 𝐷 → dom 𝐹 = 𝐷 ) |
16 |
14 15
|
sseqtrid |
⊢ ( 𝐹 : 𝐷 –1-1-onto→ 𝐷 → dom ( 𝐹 ∖ I ) ⊆ 𝐷 ) |
17 |
1 2
|
pmtrrn |
⊢ ( ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ∈ 𝑅 ) |
18 |
16 17
|
syl3an2 |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ∈ 𝑅 ) |
19 |
1 2
|
pmtrff1o |
⊢ ( ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ∈ 𝑅 → ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) : 𝐷 –1-1-onto→ 𝐷 ) |
20 |
18 19
|
syl |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) : 𝐷 –1-1-onto→ 𝐷 ) |
21 |
|
simp3 |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → dom ( 𝐹 ∖ I ) ≈ 2o ) |
22 |
1
|
pmtrmvd |
⊢ ( ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → dom ( ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ∖ I ) = dom ( 𝐹 ∖ I ) ) |
23 |
16 22
|
syl3an2 |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → dom ( ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ∖ I ) = dom ( 𝐹 ∖ I ) ) |
24 |
|
f1otrspeq |
⊢ ( ( ( 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) : 𝐷 –1-1-onto→ 𝐷 ) ∧ ( dom ( 𝐹 ∖ I ) ≈ 2o ∧ dom ( ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ∖ I ) = dom ( 𝐹 ∖ I ) ) ) → 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) |
25 |
11 20 21 23 24
|
syl22anc |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) |
26 |
25 18
|
eqeltrd |
⊢ ( ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) → 𝐹 ∈ 𝑅 ) |
27 |
10 26
|
impbii |
⊢ ( 𝐹 ∈ 𝑅 ↔ ( 𝐷 ∈ V ∧ 𝐹 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ) |