Theorem f00 5322

Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)

Assertion

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

Proof of Theorem f00

Step	Hyp	Ref	Expression
1		ffun 5283	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5289	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3408	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4764	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 133	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5134	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 414	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5250	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 121	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5286	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2175	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 304	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5321	. . 3 |- (/) : (/) --> (/)
15		feq1 5263	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5264	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 458	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F : A --> (/) <-> (/) : (/) --> (/) ) )
18	14, 17	mpbiri 167	. 2 |- ( ( F = (/) /\ A = (/) ) -> F : A --> (/) )
19	13, 18	impbii 125	1 |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   C_ wss 3076   (/)c0 3368   dom cdm 4547   ran crn 4548   Fun wfun 5125   Fn wfn 5126   -->wf 5127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-f 5135

This theorem is referenced by:  dom0  6740