Metamath Proof Explorer

Definition df-fo

Description: Define an onto function. Definition 6.15(4) of TakeutiZaring p. 27. We use their notation ("onto" under the arrow). For alternate definitions, see dffo2 , dffo3 , dffo4 , and dffo5 .

An onto function is also called a "surjection" or a "surjective function", F : A -onto-> B can be read as " F is a surjection from A onto B ". Surjections are precisely the epimorphisms in the category SetCat of sets and set functions, see setcepi . (Contributed by NM, 1-Aug-1994)

Ref Expression
Assertion df-fo ⊢ F : A ⟶ onto B ↔ F Fn A ∧ ran ⁡ F = B

Detailed syntax breakdown

Step Hyp Ref Expression
0 cF class F
1 cA class A
2 cB class B
3 1 2 0 wfo wff F : A ⟶ onto B
4 0 1 wfn wff F Fn A
5 0 crn class ran ⁡ F
6 5 2 wceq wff ran ⁡ F = B
7 4 6 wa wff F Fn A ∧ ran ⁡ F = B
8 3 7 wb wff F : A ⟶ onto B ↔ F Fn A ∧ ran ⁡ F = B