Theorem f00 4601

Description: A class is a function with empty codomain iff it and its domain are empty.

Assertion

Ref	Expression
f00	|- (F:A-->(/) <-> (F = (/) /\ A = (/)))

Proof of Theorem f00

Step	Hyp	Ref	Expression
1		df-fn 4009	. . . . 5 |- (F Fn (/) <-> (Fun F /\ dom F = (/)))
2		ffun 4565	. . . . 5 |- (F:A-->(/) -> Fun F)
3		frn 4569	. . . . . . 7 |- (F:A-->(/) -> ran F C_ (/))
4		ss0 2902	. . . . . . 7 |- (ran F C_ (/) -> ran F = (/))
5	3, 4	syl 12	. . . . . 6 |- (F:A-->(/) -> ran F = (/))
6		dm0rn0 4175	. . . . . 6 |- (dom F = (/) <-> ran F = (/))
7	5, 6	sylibr 217	. . . . 5 |- (F:A-->(/) -> dom F = (/))
8	1, 2, 7	sylanbrc 527	. . . 4 |- (F:A-->(/) -> F Fn (/))
9		fn0 4532	. . . 4 |- (F Fn (/) <-> F = (/))
10	8, 9	sylib 215	. . 3 |- (F:A-->(/) -> F = (/))
11		fdm 4567	. . . 4 |- (F:A-->(/) -> dom F = A)
12	11, 7	eqtr3d 1927	. . 3 |- (F:A-->(/) -> A = (/))
13	10, 12	jca 310	. 2 |- (F:A-->(/) -> (F = (/) /\ A = (/)))
14		f0 4600	. . 3 |- (/):(/)-->(/)
15		feq1 4551	. . . 4 |- (F = (/) -> (F:A-->(/) <-> (/):A-->(/)))
16		feq2 4552	. . . 4 |- (A = (/) -> ((/):A-->(/) <-> (/):(/)-->(/)))
17	15, 16	sylan9bb 599	. . 3 |- ((F = (/) /\ A = (/)) -> (F:A-->(/) <-> (/):(/)-->(/)))
18	14, 17	mpbiri 211	. 2 |- ((F = (/) /\ A = (/)) -> F:A-->(/))
19	13, 18	impbii 174	1 |- (F:A-->(/) <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   C_ wss 2593  (/)c0 2875  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  ga0 9453  ismgm 10367  heiborlem42 15996

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-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

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