Theorem f0dom0 6089

Description: A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)

Assertion

Ref	Expression
f0dom0	|- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )

Proof of Theorem f0dom0

Step	Hyp	Ref	Expression
1		feq2 6027	. . . 4 |- ( X = (/) -> ( F : X --> Y <-> F : (/) --> Y ) )
2		f0bi 6088	. . . . 5 |- ( F : (/) --> Y <-> F = (/) )
3	2	biimpi 206	. . . 4 |- ( F : (/) --> Y -> F = (/) )
4	1, 3	syl6bi 243	. . 3 |- ( X = (/) -> ( F : X --> Y -> F = (/) ) )
5	4	com12 32	. 2 |- ( F : X --> Y -> ( X = (/) -> F = (/) ) )
6		feq1 6026	. . . 4 |- ( F = (/) -> ( F : X --> Y <-> (/) : X --> Y ) )
7		dm0 5339	. . . . 5 |- dom (/) = (/)
8		fdm 6051	. . . . 5 |- ( (/) : X --> Y -> dom (/) = X )
9	7, 8	syl5reqr 2671	. . . 4 |- ( (/) : X --> Y -> X = (/) )
10	6, 9	syl6bi 243	. . 3 |- ( F = (/) -> ( F : X --> Y -> X = (/) ) )
11	10	com12 32	. 2 |- ( F : X --> Y -> ( F = (/) -> X = (/) ) )
12	5, 11	impbid 202	1 |- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   (/)c0 3915   dom cdm 5114   -->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:  swrdn0  13430  elfrlmbasn0  20106  mavmulsolcl  20357