HOLE Home Higher-Order Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HOLE Home  >  Th. List  >  df-f11 Unicode version

Definition df-f11 194
Description: Define a one-to-one function. (Contributed by Mario Carneiro, 10-Oct-2014.)
Assertion
Ref Expression
df-f11 |- T. |= [1-1 = \f:(al -> be) (A.\x:al (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]))]
Distinct variable group:   x,f,y

Detailed syntax breakdown of Definition df-f11
StepHypRef Expression
1 kt 8 . 2 term T.
2 tf11 189 . . 3 term 1-1
3 hal . . . . 5 type al
4 hbe . . . . 5 type be
53, 4ht 2 . . . 4 type (al -> be)
6 vf . . . 4 var f
7 tal 122 . . . . 5 term A.
8 vx . . . . . 6 var x
9 vy . . . . . . . 8 var y
105, 6tv 1 . . . . . . . . . . 11 term f:(al -> be)
113, 8tv 1 . . . . . . . . . . 11 term x:al
1210, 11kc 5 . . . . . . . . . 10 term (f:(al -> be)x:al)
133, 9tv 1 . . . . . . . . . . 11 term y:al
1410, 13kc 5 . . . . . . . . . 10 term (f:(al -> be)y:al)
15 ke 7 . . . . . . . . . 10 term =
1612, 14, 15kbr 9 . . . . . . . . 9 term [(f:(al -> be)x:al) = (f:(al -> be)y:al)]
1711, 13, 15kbr 9 . . . . . . . . 9 term [x:al = y:al]
18 tim 121 . . . . . . . . 9 term ==>
1916, 17, 18kbr 9 . . . . . . . 8 term [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]
203, 9, 19kl 6 . . . . . . 7 term \y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]
217, 20kc 5 . . . . . 6 term (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]])
223, 8, 21kl 6 . . . . 5 term \x:al (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]])
237, 22kc 5 . . . 4 term (A.\x:al (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]))
245, 6, 23kl 6 . . 3 term \f:(al -> be) (A.\x:al (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]))
252, 24, 15kbr 9 . 2 term [1-1 = \f:(al -> be) (A.\x:al (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]))]
261, 25wffMMJ2 11 1 wff T. |= [1-1 = \f:(al -> be) (A.\x:al (A.\y:al [[(f:(al -> be)x:al) = (f:(al -> be)y:al)] ==> [x:al = y:al]]))]
Colors of variables: type var term
This definition is referenced by: (None)
  Copyright terms: Public domain W3C validator