Theorem foconst 3689

Description: A non-zero constant function is onto.

Assertion

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

Proof of Theorem foconst

Step	Hyp	Ref	Expression
1		frel 3636	. . . . 5 |- (F:A-->{B} -> Rel F)
2		relrn0 3362	. . . . . 6 |- (Rel F -> (F = (/) <-> ran F = (/)))
3	2	necon3abid 1602	. . . . 5 |- (Rel F -> (F =/= (/) <-> -. ran F = (/)))
4	1, 3	syl 10	. . . 4 |- (F:A-->{B} -> (F =/= (/) <-> -. ran F = (/)))
5		frn 3639	. . . . . 6 |- (F:A-->{B} -> ran F (_ {B})
6		sssn 2477	. . . . . 6 |- (ran F (_ {B} <-> (ran F = (/) \/ ran F = {B}))
7	5, 6	sylib 198	. . . . 5 |- (F:A-->{B} -> (ran F = (/) \/ ran F = {B}))
8	7	ord 232	. . . 4 |- (F:A-->{B} -> (-. ran F = (/) -> ran F = {B}))
9	4, 8	sylbid 203	. . 3 |- (F:A-->{B} -> (F =/= (/) -> ran F = {B}))
10	9	imdistani 445	. 2 |- ((F:A-->{B} /\ F =/= (/)) -> (F:A-->{B} /\ ran F = {B}))
11		dffo2 3681	. 2 |- (F:A-onto->{B} <-> (F:A-->{B} /\ ran F = {B}))
12	10, 11	sylibr 200	1 |- ((F:A-->{B} /\ F =/= (/)) -> F:A-onto->{B})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   =/= wne 1588   (_ wss 2050  (/)c0 2283  {csn 2413  ran crn 3177  Rel wrel 3181  -->wf 3184  -onto->wfo 3186

This theorem is referenced by:  cnconst 7777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202

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