Theorem f00 5379

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 5340	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5346	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3449	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4821	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 133	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5191	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 414	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5307	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 121	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5343	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2200	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 304	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5378	. . 3 |- (/) : (/) --> (/)
15		feq1 5320	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5321	. . . 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 1343   C_ wss 3116   (/)c0 3409   dom cdm 4604   ran crn 4605   Fun wfun 5182   Fn wfn 5183   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  dom0  6804