Theorem dff1o3 5482

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 991	. 2 |- ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ ran F = B ) /\ Fun `' F ) )
2		dff1o2 5481	. 2 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
3		df-fo 5237	. . 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 980   = wceq 1364   `'ccnv 4640   ran crn 4642   Fun wfun 5225   Fn wfn 5226   -onto->wfo 5229   -1-1-onto->wf1o 5230

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238

This theorem is referenced by:  f1ofo  5483  resdif  5498  f11o  5509  f1opw  6096  1stconst  6240  2ndconst  6241  f1o2ndf1  6247  ssdomg  6796  phplem4  6873  phplem4on  6885