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	$⊢ F : A \underset{onto}{⟶} B \leftrightarrow (F Fn A \land ran F = B)$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cF	$class F$
1		cA	$class A$
2		cB	$class B$
3	1 2 0	wfo	$wff F : A \underset{onto}{⟶} B$
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	$wff (F Fn A \land ran F = B)$
8	3 7	wb	$wff (F : A \underset{onto}{⟶} B \leftrightarrow (F Fn A \land ran F = B))$