Theorem f00 3657

Description: A class is a function with empty codomain iff it and its domain are empty.

Assertion

Ref	Expression
f00	|- (F:A-->(/) <-> (F = (/) /\ A = (/)))

Proof of Theorem f00

Step	Hyp	Ref	Expression
1		ffun 3629	. . . . . 6 |- (F:A-->(/) -> Fun F)
2		frn 3633	. . . . . . . 8 |- (F:A-->(/) -> ran F (_ (/))
3		ss0 2303	. . . . . . . 8 |- (ran F (_ (/) -> ran F = (/))
4	2, 3	syl 10	. . . . . . 7 |- (F:A-->(/) -> ran F = (/))
5		dm0rn0 3330	. . . . . . 7 |- (dom F = (/) <-> ran F = (/))
6	4, 5	sylibr 200	. . . . . 6 |- (F:A-->(/) -> dom F = (/))
7	1, 6	jca 288	. . . . 5 |- (F:A-->(/) -> (Fun F /\ dom F = (/)))
8		df-fn 3193	. . . . 5 |- (F Fn (/) <-> (Fun F /\ dom F = (/)))
9	7, 8	sylibr 200	. . . 4 |- (F:A-->(/) -> F Fn (/))
10		fn0 3605	. . . 4 |- (F Fn (/) <-> F = (/))
11	9, 10	sylib 198	. . 3 |- (F:A-->(/) -> F = (/))
12		fdm 3631	. . . 4 |- (F:A-->(/) -> dom F = A)
13	12, 6	eqtr3d 1509	. . 3 |- (F:A-->(/) -> A = (/))
14	11, 13	jca 288	. 2 |- (F:A-->(/) -> (F = (/) /\ A = (/)))
15		f0 3656	. . 3 |- (/):(/)-->(/)
16		feq1 3620	. . . 4 |- (F = (/) -> (F:A-->(/) <-> (/):A-->(/)))
17		feq2 3621	. . . 4 |- (A = (/) -> ((/):A-->(/) <-> (/):(/)-->(/)))
18	16, 17	sylan9bb 540	. . 3 |- ((F = (/) /\ A = (/)) -> (F:A-->(/) <-> (/):(/)-->(/)))
19	15, 18	mpbiri 194	. 2 |- ((F = (/) /\ A = (/)) -> F:A-->(/))
20	14, 19	impbi 157	1 |- (F:A-->(/) <-> (F = (/) /\ A = (/)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   (_ wss 2047  (/)c0 2280  dom cdm 3170  ran crn 3171  Fun wfun 3176   Fn wfn 3177  -->wf 3178

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

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