Theorem fn0 4961

Description: A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fn0	|- ( F Fn (/) <-> F = (/))

Proof of Theorem fn0

Step	Hyp	Ref	Expression
1		fnrel 4940	. . 3 |- ( F Fn (/) -> Rel F)
2		fndm 4941	. . 3 |- ( F Fn (/) -> dom F = (/))
3		reldm0 4496	. . . 4 |- ( Rel F -> ( F = (/) <-> dom F = (/)))
4	3	biimpar 281	. . 3 |- (( Rel F /\ dom F = (/)) -> F = (/))
5	1, 2, 4	syl2anc 391	. 2 |- ( F Fn (/) -> F = (/))
6		fun0 4900	. . . 4 |- Fun (/)
7		dm0 4492	. . . 4 |- dom (/) = (/)
8		df-fn 4848	. . . 4 |- ( (/) Fn (/) <-> ( Fun (/) /\ dom (/) = (/)))
9	6, 7, 8	mpbir2an 848	. . 3 |- (/) Fn (/)
10		fneq1 4930	. . 3 |- ( F = (/) -> ( F Fn (/) <-> (/) Fn (/)))
11	9, 10	mpbiri 157	. 2 |- ( F = (/) -> F Fn (/))
12	5, 11	impbii 117	1 |- ( F Fn (/) <-> F = (/))

Syntax hints:   <-> wb 98   = wceq 1242   (/)c0 3218   dom cdm 4288   Rel wrel 4293   Fun wfun 4839   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-fun 4847  df-fn 4848

This theorem is referenced by:  mpt0  4969  f0  5023  f00  5024  f1o00  5104  fo00  5105  tpos0  5830  0fz1  8659