Theorem f00 6087

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 6048	. . . . 5 |- ( F : A --> (/) -> Fun F )
2		frn 6053	. . . . . . 7 |- ( F : A --> (/) -> ran F C_ (/) )
3		ss0 3974	. . . . . . 7 |- ( ran F C_ (/) -> ran F = (/) )
4	2, 3	syl 17	. . . . . 6 |- ( F : A --> (/) -> ran F = (/) )
5		dm0rn0 5342	. . . . . 6 |- ( dom F = (/) <-> ran F = (/) )
6	4, 5	sylibr 224	. . . . 5 |- ( F : A --> (/) -> dom F = (/) )
7		df-fn 5891	. . . . 5 |- ( F Fn (/) <-> ( Fun F /\ dom F = (/) ) )
8	1, 6, 7	sylanbrc 698	. . . 4 |- ( F : A --> (/) -> F Fn (/) )
9		fn0 6011	. . . 4 |- ( F Fn (/) <-> F = (/) )
10	8, 9	sylib 208	. . 3 |- ( F : A --> (/) -> F = (/) )
11		fdm 6051	. . . 4 |- ( F : A --> (/) -> dom F = A )
12	11, 6	eqtr3d 2658	. . 3 |- ( F : A --> (/) -> A = (/) )
13	10, 12	jca 554	. 2 |- ( F : A --> (/) -> ( F = (/) /\ A = (/) ) )
14		f0 6086	. . 3 |- (/) : (/) --> (/)
15		feq1 6026	. . . 4 |- ( F = (/) -> ( F : A --> (/) <-> (/) : A --> (/) ) )
16		feq2 6027	. . . 4 |- ( A = (/) -> ( (/) : A --> (/) <-> (/) : (/) --> (/) ) )
17	15, 16	sylan9bb 736	. . 3 |- ( ( F = (/) /\ A = (/) ) -> ( F : A --> (/) <-> (/) : (/) --> (/) ) )
18	14, 17	mpbiri 248	. 2 |- ( ( F = (/) /\ A = (/) ) -> F : A --> (/) )
19	13, 18	impbii 199	1 |- ( F : A --> (/) <-> ( F = (/) /\ A = (/) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   (/)c0 3915   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  cantnff  8571  0wrd0  13331  supcvg  14588  ram0  15726  itgsubstlem  23811  uhgr0vb  25967  lfuhgr1v0e  26146  wlkv0  26547  ismgmOLD  33649