Theorem f00 5081

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 5048	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5052	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3257	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4552	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 137	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 4905	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 394	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5018	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 127	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5050	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2074	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 290	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5080	. . 3 |- (/) : (/) --> (/)
15		feq1 5030	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5031	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 435	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F : A --> (/) <-> (/) : (/) --> (/) ) )
18	14, 17	mpbiri 157	. 2 |- ( ( F = (/) /\ A = (/) ) -> F : A --> (/) )
19	13, 18	impbii 117	1 |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   C_ wss 2917   (/)c0 3224   dom cdm 4345   ran crn 4346   Fun wfun 4896   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)