Theorem foconst 6126

Description: A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)

Assertion

Ref	Expression
foconst	|- ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } )

Proof of Theorem foconst

Step	Hyp	Ref	Expression
1		frel 6050	. . . . 5 |- ( F : A --> { B } -> Rel F )
2		relrn0 5383	. . . . . 6 |- ( Rel F -> ( F = (/) <-> ran F = (/) ) )
3	2	necon3abid 2830	. . . . 5 |- ( Rel F -> ( F =/= (/) <-> -. ran F = (/) ) )
4	1, 3	syl 17	. . . 4 |- ( F : A --> { B } -> ( F =/= (/) <-> -. ran F = (/) ) )
5		frn 6053	. . . . . 6 |- ( F : A --> { B } -> ran F C_ { B } )
6		sssn 4358	. . . . . 6 |- ( ran F C_ { B } <-> ( ran F = (/) \/ ran F = { B } ) )
7	5, 6	sylib 208	. . . . 5 |- ( F : A --> { B } -> ( ran F = (/) \/ ran F = { B } ) )
8	7	ord 392	. . . 4 |- ( F : A --> { B } -> ( -. ran F = (/) -> ran F = { B } ) )
9	4, 8	sylbid 230	. . 3 |- ( F : A --> { B } -> ( F =/= (/) -> ran F = { B } ) )
10	9	imdistani 726	. 2 |- ( ( F : A --> { B } /\ F =/= (/) ) -> ( F : A --> { B } /\ ran F = { B } ) )
11		dffo2 6119	. 2 |- ( F : A -onto-> { B } <-> ( F : A --> { B } /\ ran F = { B } ) )
12	10, 11	sylibr 224	1 |- ( ( F : A --> { B } /\ F =/= (/) ) -> F : A -onto-> { B } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   C_ wss 3574   (/)c0 3915   {csn 4177   ran crn 5115   Rel wrel 5119   -->wf 5884   -onto->wfo 5886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894

This theorem is referenced by: (None)