Metamath Proof Explorer


Definition df-tx

Description: Define the binary topological product, which is homeomorphic to the general topological product over a two element set, but is more convenient to use. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion df-tx
|- tX = ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 ctx
 |-  tX
1 vr
 |-  r
2 cvv
 |-  _V
3 vs
 |-  s
4 ctg
 |-  topGen
5 vx
 |-  x
6 1 cv
 |-  r
7 vy
 |-  y
8 3 cv
 |-  s
9 5 cv
 |-  x
10 7 cv
 |-  y
11 9 10 cxp
 |-  ( x X. y )
12 5 7 6 8 11 cmpo
 |-  ( x e. r , y e. s |-> ( x X. y ) )
13 12 crn
 |-  ran ( x e. r , y e. s |-> ( x X. y ) )
14 13 4 cfv
 |-  ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) )
15 1 3 2 2 14 cmpo
 |-  ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )
16 0 15 wceq
 |-  tX = ( r e. _V , s e. _V |-> ( topGen ` ran ( x e. r , y e. s |-> ( x X. y ) ) ) )