Theorem dff1o3 5448

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o3	|- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )

Proof of Theorem dff1o3

Step	Hyp	Ref	Expression
1		3anan32 984	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
2		dff1o2 5447	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
3		df-fo 5204	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	3	anbi1i 455	. 2 |- ( ( F : A -onto-> B /\ Fun `' F ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
5	1, 2, 4	3bitr4i 211	1 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   `'ccnv 4610   ran crn 4612   Fun wfun 5192   Fn wfn 5193   -onto->wfo 5196   -1-1-onto->wf1o 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1ofo  5449  resdif  5464  f11o  5475  f1opw  6056  1stconst  6200  2ndconst  6201  f1o2ndf1  6207  ssdomg  6756  phplem4  6833  phplem4on  6845