Theorem f0dom0 5530

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 5466	. . . 4 |- ( X = (/) -> ( F : X --> Y <-> F : (/) --> Y ) )
2		f0bi 5529	. . . . 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 5465	. . . 4 |- ( F = (/) -> ( F : X --> Y <-> (/) : X --> Y ) )
7		fdm 5488	. . . . 5 |- ( (/) : X --> Y -> dom (/) = X )
8		dm0 4945	. . . . 5 |- dom (/) = (/)
9	7, 8	eqtr3di 2279	. . . 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 1397   (/)c0 3494   dom cdm 4725   -->wf 5322

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  pfxn0  11268