Theorem dffo2 3660

Description: Alternate definition of an onto function.

Assertion

Ref	Expression
dffo2	|- (F:A-onto->B <-> (F:A-->B /\ ran F = B))

Proof of Theorem dffo2

Step	Hyp	Ref	Expression
1		fof 3657	. . 3 |- (F:A-onto->B -> F:A-->B)
2		forn 3659	. . 3 |- (F:A-onto->B -> ran F = B)
3	1, 2	jca 288	. 2 |- (F:A-onto->B -> (F:A-->B /\ ran F = B))
4		df-fo 3186	. . . 4 |- (F:A-onto->B <-> (F Fn A /\ ran F = B))
5	4	biimpr 152	. . 3 |- ((F Fn A /\ ran F = B) -> F:A-onto->B)
6		ffn 3613	. . 3 |- (F:A-->B -> F Fn A)
7	5, 6	sylan 448	. 2 |- ((F:A-->B /\ ran F = B) -> F:A-onto->B)
8	3, 7	impbi 157	1 |- (F:A-onto->B <-> (F:A-->B /\ ran F = B))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953  ran crn 3161   Fn wfn 3167  -->wf 3168  -onto->wfo 3170

This theorem is referenced by:  foco 3667  foconst 3668  dffo3 3804  dffo4 3805  exfo 3807  2ndconst 4081  grpfo 7977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043  df-f 3184  df-fo 3186

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