Theorem f00 5517

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 5476	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5482	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3532	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4940	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 134	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5321	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 417	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5443	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 122	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5479	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2264	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 306	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5516	. . 3 |- (/) : (/) --> (/)
15		feq1 5456	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5457	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 462	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F : A --> (/) <-> (/) : (/) --> (/) ) )
18	14, 17	mpbiri 168	. 2 |- ( ( F = (/) /\ A = (/) ) -> F : A --> (/) )
19	13, 18	impbii 126	1 |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   C_ wss 3197   (/)c0 3491   dom cdm 4719   ran crn 4720   Fun wfun 5312   Fn wfn 5313   -->wf 5314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322

This theorem is referenced by:  dom0  6999  0wrd0  11097  uhgr0vb  15884  wlkv0  16080