Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrfval.t |
⊢ 𝑇 = ( pmTrsp ‘ 𝐷 ) |
2 |
|
simpl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝐷 ∈ 𝑉 ) |
3 |
|
simpr1 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 ∈ 𝐷 ) |
4 |
|
simpr2 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ∈ 𝐷 ) |
5 |
3 4
|
prssd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → { 𝑋 , 𝑌 } ⊆ 𝐷 ) |
6 |
|
pr2nelem |
⊢ ( ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ≈ 2o ) |
7 |
6
|
adantl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → { 𝑋 , 𝑌 } ≈ 2o ) |
8 |
1
|
pmtrfv |
⊢ ( ( ( 𝐷 ∈ 𝑉 ∧ { 𝑋 , 𝑌 } ⊆ 𝐷 ∧ { 𝑋 , 𝑌 } ≈ 2o ) ∧ 𝑋 ∈ 𝐷 ) → ( ( 𝑇 ‘ { 𝑋 , 𝑌 } ) ‘ 𝑋 ) = if ( 𝑋 ∈ { 𝑋 , 𝑌 } , ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) , 𝑋 ) ) |
9 |
2 5 7 3 8
|
syl31anc |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑇 ‘ { 𝑋 , 𝑌 } ) ‘ 𝑋 ) = if ( 𝑋 ∈ { 𝑋 , 𝑌 } , ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) , 𝑋 ) ) |
10 |
|
prid1g |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 ∈ { 𝑋 , 𝑌 } ) |
11 |
3 10
|
syl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 ∈ { 𝑋 , 𝑌 } ) |
12 |
11
|
iftrued |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → if ( 𝑋 ∈ { 𝑋 , 𝑌 } , ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) , 𝑋 ) = ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) ) |
13 |
|
difprsnss |
⊢ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) ⊆ { 𝑌 } |
14 |
13
|
a1i |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) ⊆ { 𝑌 } ) |
15 |
|
prid2g |
⊢ ( 𝑌 ∈ 𝐷 → 𝑌 ∈ { 𝑋 , 𝑌 } ) |
16 |
4 15
|
syl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ∈ { 𝑋 , 𝑌 } ) |
17 |
|
simpr3 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 ≠ 𝑌 ) |
18 |
17
|
necomd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ≠ 𝑋 ) |
19 |
|
eldifsn |
⊢ ( 𝑌 ∈ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) ↔ ( 𝑌 ∈ { 𝑋 , 𝑌 } ∧ 𝑌 ≠ 𝑋 ) ) |
20 |
16 18 19
|
sylanbrc |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑌 ∈ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) ) |
21 |
20
|
snssd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → { 𝑌 } ⊆ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) ) |
22 |
14 21
|
eqssd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) = { 𝑌 } ) |
23 |
22
|
unieqd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) = ∪ { 𝑌 } ) |
24 |
|
unisng |
⊢ ( 𝑌 ∈ 𝐷 → ∪ { 𝑌 } = 𝑌 ) |
25 |
4 24
|
syl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ∪ { 𝑌 } = 𝑌 ) |
26 |
23 25
|
eqtrd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) = 𝑌 ) |
27 |
12 26
|
eqtrd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → if ( 𝑋 ∈ { 𝑋 , 𝑌 } , ∪ ( { 𝑋 , 𝑌 } ∖ { 𝑋 } ) , 𝑋 ) = 𝑌 ) |
28 |
9 27
|
eqtrd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ ( 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌 ) ) → ( ( 𝑇 ‘ { 𝑋 , 𝑌 } ) ‘ 𝑋 ) = 𝑌 ) |