Theorem dff1o5 4646

Description: Alternate definition of one-to-one onto function. (The proof was 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 4013	. 2 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
2		f1f 4610	. . . . 5 |- (F:A-1-1->B -> F:A-->B)
3		ibar 705	. . . . 5 |- (F:A-->B -> (ran F = B <-> (F:A-->B /\ ran F = B)))
4	2, 3	syl 12	. . . 4 |- (F:A-1-1->B -> (ran F = B <-> (F:A-->B /\ ran F = B)))
5		dffo2 4621	. . . 4 |- (F:A-onto->B <-> (F:A-->B /\ ran F = B))
6	4, 5	syl6rbbr 598	. . 3 |- (F:A-1-1->B -> (F:A-onto->B <-> ran F = B))
7	6	pm5.32i 707	. 2 |- ((F:A-1-1->B /\ F:A-onto->B) <-> (F:A-1-1->B /\ ran F = B))
8	1, 7	bitri 190	1 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298  ran crn 3987  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996  -1-1-onto->wf1o 3997

This theorem is referenced by:  f1orescnv 4655  ac6sfi 5509  mapenlem2 5584  om2uzf1oi 7712  grplactf1o 9406  logrn 10105  hmeogrp 14892  homcard 14893  enf1f1o 15720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013

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