Theorem f0dom0 5421

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 5361	. . . 4 |- ( X = (/) -> ( F : X --> Y <-> F : (/) --> Y ) )
2		f0bi 5420	. . . . 5 |- ( F : (/) --> Y <-> F = (/) )
3	2	biimpi 120	. . . 4 |- ( F : (/) --> Y -> F = (/) )
4	1, 3	biimtrdi 163	. . 3 |- ( X = (/) -> ( F : X --> Y -> F = (/) ) )
5	4	com12 30	. 2 |- ( F : X --> Y -> ( X = (/) -> F = (/) ) )
6		feq1 5360	. . . 4 |- ( F = (/) -> ( F : X --> Y <-> (/) : X --> Y ) )
7		fdm 5383	. . . . 5 |- ( (/) : X --> Y -> dom (/) = X )
8		dm0 4853	. . . . 5 |- dom (/) = (/)
9	7, 8	eqtr3di 2235	. . . 4 |- ( (/) : X --> Y -> X = (/) )
10	6, 9	biimtrdi 163	. . 3 |- ( F = (/) -> ( F : X --> Y -> X = (/) ) )
11	10	com12 30	. 2 |- ( F : X --> Y -> ( F = (/) -> X = (/) ) )
12	5, 11	impbid 129	1 |- ( F : X --> Y -> ( X = (/) <-> F = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   (/)c0 3434   dom cdm 4638   -->wf 5224

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232

This theorem is referenced by: (None)