Theorem dff1o5 5623

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o5	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )

Proof of Theorem dff1o5

Step	Hyp	Ref	Expression
1		df-f1o 5359	. 2 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dffo2 5594	. . . 4 |- ( F : A -onto-> B <-> ( F : A --> B /\ ran F = B ) )
3		f1f 5573	. . . . 5 |- ( F : A -1-1-> B -> F : A --> B )
4	3	biantrurd 305	. . . 4 |- ( F : A -1-1-> B -> ( ran F = B <-> ( F : A --> B /\ ran F = B ) ) )
5	2, 4	bitr4id 199	. . 3 |- ( F : A -1-1-> B -> ( F : A -onto-> B <-> ran F = B ) )
6	5	pm5.32i 454	. 2 |- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F : A -1-1-> B /\ ran F = B ) )
7	1, 6	bitri 184	1 |- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ ran F = B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   ran crn 4750   -->wf 5348   -1-1->wf1 5349   -onto->wfo 5350   -1-1-onto->wf1o 5351

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359

This theorem is referenced by:  f1orescnv  5630  f1finf1o  7217  djuinr  7354  eninl  7388  eninr  7389  frec2uzf1od  10768  ennnfonelemex  13165  ennnfonelemen  13172  ssnnctlemct  13197  2lgslem1b  15962  ausgrusgrben  16163  pwf1oexmid  16773