Theorem f00 5452

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 5413	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5419	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3492	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4884	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 134	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5262	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 417	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5380	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 122	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5416	. . . 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 5451	. . 3 |- (/) : (/) --> (/)
15		feq1 5393	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5394	. . . 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 3451   dom cdm 4664   ran crn 4665   Fun wfun 5253   Fn wfn 5254   -->wf 5255

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 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-fn 5262  df-f 5263

This theorem is referenced by:  dom0  6908  0wrd0  10978