Step |
Hyp |
Ref |
Expression |
1 |
|
xmetsym |
β’ ( ( π· β ( βMet β π ) β§ π₯ β π β§ π¦ β π ) β ( π₯ π· π¦ ) = ( π¦ π· π₯ ) ) |
2 |
1
|
3expb |
β’ ( ( π· β ( βMet β π ) β§ ( π₯ β π β§ π¦ β π ) ) β ( π₯ π· π¦ ) = ( π¦ π· π₯ ) ) |
3 |
2
|
ralrimivva |
β’ ( π· β ( βMet β π ) β β π₯ β π β π¦ β π ( π₯ π· π¦ ) = ( π¦ π· π₯ ) ) |
4 |
|
xmetf |
β’ ( π· β ( βMet β π ) β π· : ( π Γ π ) βΆ β* ) |
5 |
|
ffn |
β’ ( π· : ( π Γ π ) βΆ β* β π· Fn ( π Γ π ) ) |
6 |
|
tpossym |
β’ ( π· Fn ( π Γ π ) β ( tpos π· = π· β β π₯ β π β π¦ β π ( π₯ π· π¦ ) = ( π¦ π· π₯ ) ) ) |
7 |
4 5 6
|
3syl |
β’ ( π· β ( βMet β π ) β ( tpos π· = π· β β π₯ β π β π¦ β π ( π₯ π· π¦ ) = ( π¦ π· π₯ ) ) ) |
8 |
3 7
|
mpbird |
β’ ( π· β ( βMet β π ) β tpos π· = π· ) |