Theorem f00 5537

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 5492	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5498	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3537	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4954	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 134	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5336	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 417	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5459	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 122	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5495	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2266	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 306	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5536	. . 3 |- (/) : (/) --> (/)
15		feq1 5472	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5473	. . . 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 1398   C_ wss 3201   (/)c0 3496   dom cdm 4731   ran crn 4732   Fun wfun 5327   Fn wfn 5328   -->wf 5329

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  dom0  7067  0wrd0  11205  uhgr0vb  16025  wlkv0  16310  gfsumval  16809