Theorem f00 5449

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 5410	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5416	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3491	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4883	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 134	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5261	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 417	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5377	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 122	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5413	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2231	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 306	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5448	. . 3 |- (/) : (/) --> (/)
15		feq1 5390	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5391	. . . 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 1364   C_ wss 3157   (/)c0 3450   dom cdm 4663   ran crn 4664   Fun wfun 5252   Fn wfn 5253   -->wf 5254

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-fun 5260  df-fn 5261  df-f 5262

This theorem is referenced by:  dom0  6899  0wrd0  10961