Theorem f00 5314

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 5275	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5281	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3403	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4756	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 133	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5126	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 413	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5242	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 121	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5278	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2174	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 304	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5313	. . 3 |- (/) : (/) --> (/)
15		feq1 5255	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5256	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 457	. . 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 1331   C_ wss 3071   (/)c0 3363   dom cdm 4539   ran crn 4540   Fun wfun 5117   Fn wfn 5118   -->wf 5119

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127

This theorem is referenced by:  dom0  6732