Theorem f00 3648

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 3621	. . . . . 6 |- (F:A-->(/) -> Fun F)
2		frn 3624	. . . . . . . 8 |- (F:A-->(/) -> ran F (_ (/))
3		ss0 2299	. . . . . . . 8 |- (ran F (_ (/) -> ran F = (/))
4	2, 3	syl 10	. . . . . . 7 |- (F:A-->(/) -> ran F = (/))
5		dm0rn0 3325	. . . . . . 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 3188	. . . . 5 |- (F Fn (/) <-> (Fun F /\ dom F = (/)))
9	7, 8	sylibr 200	. . . 4 |- (F:A-->(/) -> F Fn (/))
10		fn0 3597	. . . 4 |- (F Fn (/) <-> F = (/))
11	9, 10	sylib 198	. . 3 |- (F:A-->(/) -> F = (/))
12		fdm 3623	. . . 4 |- (F:A-->(/) -> dom F = A)
13	12, 6	eqtr3d 1506	. . 3 |- (F:A-->(/) -> A = (/))
14	11, 13	jca 288	. 2 |- (F:A-->(/) -> (F = (/) /\ A = (/)))
15		f0 3647	. . 3 |- (/):(/)-->(/)
16		feq1 3612	. . . 4 |- (F = (/) -> (F:A-->(/) <-> (/):A-->(/)))
17		feq2 3613	. . . 4 |- (A = (/) -> ((/):A-->(/) <-> (/):(/)-->(/)))
18	16, 17	sylan9bb 539	. . 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 954   (_ wss 2043  (/)c0 2276  dom cdm 3165  ran crn 3166  Fun wfun 3171   Fn wfn 3172  -->wf 3173

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189

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