Step |
Hyp |
Ref |
Expression |
1 |
|
pmtridf1o.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
pmtridf1o.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
3 |
|
pmtridf1o.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐴 ) |
4 |
|
pmtridf1o.t |
⊢ 𝑇 = if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) |
5 |
|
fvresi |
⊢ ( 𝑌 ∈ 𝐴 → ( ( I ↾ 𝐴 ) ‘ 𝑌 ) = 𝑌 ) |
6 |
3 5
|
syl |
⊢ ( 𝜑 → ( ( I ↾ 𝐴 ) ‘ 𝑌 ) = 𝑌 ) |
7 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( ( I ↾ 𝐴 ) ‘ 𝑌 ) = 𝑌 ) |
8 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑋 = 𝑌 ) |
9 |
8
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) = ( I ↾ 𝐴 ) ) |
10 |
4 9
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑇 = ( I ↾ 𝐴 ) ) |
11 |
10
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( 𝑇 ‘ 𝑌 ) = ( ( I ↾ 𝐴 ) ‘ 𝑌 ) ) |
12 |
7 11 8
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( 𝑇 ‘ 𝑌 ) = 𝑋 ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑋 ≠ 𝑌 ) |
14 |
13
|
neneqd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ¬ 𝑋 = 𝑌 ) |
15 |
14
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) = ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) |
16 |
4 15
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑇 = ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) |
17 |
16
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑇 ‘ 𝑌 ) = ( ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ‘ 𝑌 ) ) |
18 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝐴 ∈ 𝑉 ) |
19 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑋 ∈ 𝐴 ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑌 ∈ 𝐴 ) |
21 |
|
eqid |
⊢ ( pmTrsp ‘ 𝐴 ) = ( pmTrsp ‘ 𝐴 ) |
22 |
21
|
pmtrprfv2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ‘ 𝑌 ) = 𝑋 ) |
23 |
18 19 20 13 22
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ‘ 𝑌 ) = 𝑋 ) |
24 |
17 23
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑇 ‘ 𝑌 ) = 𝑋 ) |
25 |
12 24
|
pm2.61dane |
⊢ ( 𝜑 → ( 𝑇 ‘ 𝑌 ) = 𝑋 ) |