Theorem f0dom0 5581

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 5512	. . . 4 |- ( X = (/) -> ( F : X --> Y <-> F : (/) --> Y ) )
2		f0bi 5580	. . . . 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 5511	. . . 4 |- ( F = (/) -> ( F : X --> Y <-> (/) : X --> Y ) )
7		fdm 5534	. . . . 5 |- ( (/) : X --> Y -> dom (/) = X )
8		dm0 4990	. . . . 5 |- dom (/) = (/)
9	7, 8	eqtr3di 2286	. . . 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 1402   (/)c0 3520   dom cdm 4769   -->wf 5368

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  pfxn0  11438