Theorem dff1o3 5530

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 992	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
2		dff1o2 5529	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
3		df-fo 5278	. . 3 |- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	3	anbi1i 458	. 2 |- ( ( F : A -onto-> B /\ Fun `' F ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
5	1, 2, 4	3bitr4i 212	1 |- ( F : A -1-1-onto-> B <-> ( F : A -onto-> B /\ Fun `' F ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   `'ccnv 4675   ran crn 4677   Fun wfun 5266   Fn wfn 5267   -onto->wfo 5270   -1-1-onto->wf1o 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279

This theorem is referenced by:  f1ofo  5531  resdif  5546  f11o  5557  f1opw  6155  1stconst  6309  2ndconst  6310  f1o2ndf1  6316  ssdomg  6872  phplem4  6954  phplem4on  6966