Theorem f00 5389

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 5350	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 5356	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3455	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 14	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 4828	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 133	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5201	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 415	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 5317	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 121	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 5353	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2205	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 304	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 5388	. . 3 |- (/) : (/) --> (/)
15		feq1 5330	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 5331	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 459	. . 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 1348   C_ wss 3121   (/)c0 3414   dom cdm 4611   ran crn 4612   Fun wfun 5192   Fn wfn 5193   -->wf 5194

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  dom0  6816