Theorem f00 5584

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 5536	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5542	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3563	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4998	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 134	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5380	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 421	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5503	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 122	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5539	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2273	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 306	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5583	. . 3 |- (/) : (/) --> (/)
15		feq1 5516	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5517	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 466	. . 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 = (/) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   C_ wss 3220   (/)c0 3520   dom cdm 4774   ran crn 4775   Fun wfun 5371   Fn wfn 5372   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381

This theorem is used by:  dom0  7138  0wrd0  11330  gsumvalfi  14152  uhgr0vb  16325  wlkv0  16610