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 |
|
iftrue |
⊢ ( 𝑋 = 𝑌 → if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) = ( I ↾ 𝐴 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) = ( I ↾ 𝐴 ) ) |
7 |
4 6
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑇 = ( I ↾ 𝐴 ) ) |
8 |
|
f1oi |
⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 |
9 |
8
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 ) |
10 |
|
f1oeq1 |
⊢ ( 𝑇 = ( I ↾ 𝐴 ) → ( 𝑇 : 𝐴 –1-1-onto→ 𝐴 ↔ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 ) ) |
11 |
10
|
biimpar |
⊢ ( ( 𝑇 = ( I ↾ 𝐴 ) ∧ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 ) → 𝑇 : 𝐴 –1-1-onto→ 𝐴 ) |
12 |
7 9 11
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑇 : 𝐴 –1-1-onto→ 𝐴 ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑋 ≠ 𝑌 ) |
14 |
13
|
neneqd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ¬ 𝑋 = 𝑌 ) |
15 |
|
iffalse |
⊢ ( ¬ 𝑋 = 𝑌 → if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) = ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) |
16 |
14 15
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → if ( 𝑋 = 𝑌 , ( I ↾ 𝐴 ) , ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) = ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ) |
17 |
4 16
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑇 = ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 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 |
19 20
|
prssd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ⊆ 𝐴 ) |
22 |
|
pr2nelem |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ≈ 2o ) |
23 |
19 20 13 22
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ≈ 2o ) |
24 |
|
eqid |
⊢ ( pmTrsp ‘ 𝐴 ) = ( pmTrsp ‘ 𝐴 ) |
25 |
|
eqid |
⊢ ran ( pmTrsp ‘ 𝐴 ) = ran ( pmTrsp ‘ 𝐴 ) |
26 |
24 25
|
pmtrrn |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ { 𝑋 , 𝑌 } ⊆ 𝐴 ∧ { 𝑋 , 𝑌 } ≈ 2o ) → ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ∈ ran ( pmTrsp ‘ 𝐴 ) ) |
27 |
18 21 23 26
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( ( pmTrsp ‘ 𝐴 ) ‘ { 𝑋 , 𝑌 } ) ∈ ran ( pmTrsp ‘ 𝐴 ) ) |
28 |
17 27
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑇 ∈ ran ( pmTrsp ‘ 𝐴 ) ) |
29 |
24 25
|
pmtrff1o |
⊢ ( 𝑇 ∈ ran ( pmTrsp ‘ 𝐴 ) → 𝑇 : 𝐴 –1-1-onto→ 𝐴 ) |
30 |
28 29
|
syl |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑇 : 𝐴 –1-1-onto→ 𝐴 ) |
31 |
12 30
|
pm2.61dane |
⊢ ( 𝜑 → 𝑇 : 𝐴 –1-1-onto→ 𝐴 ) |