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Theorem mapdm0 6487
Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.)
Assertion
Ref Expression
mapdm0  |-  ( B  e.  V  ->  ( B  ^m  (/) )  =  { (/)
} )

Proof of Theorem mapdm0
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 0ex 3995 . . . . 5  |-  (/)  e.  _V
2 elmapg 6485 . . . . 5  |-  ( ( B  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( B  ^m  (/) )  <->  f : (/) --> B ) )
31, 2mpan2 419 . . . 4  |-  ( B  e.  V  ->  (
f  e.  ( B  ^m  (/) )  <->  f : (/) --> B ) )
4 f0bi 5251 . . . 4  |-  ( f : (/) --> B  <->  f  =  (/) )
53, 4syl6bb 195 . . 3  |-  ( B  e.  V  ->  (
f  e.  ( B  ^m  (/) )  <->  f  =  (/) ) )
6 vex 2644 . . . 4  |-  f  e. 
_V
76elsn 3490 . . 3  |-  ( f  e.  { (/) }  <->  f  =  (/) )
85, 7syl6bbr 197 . 2  |-  ( B  e.  V  ->  (
f  e.  ( B  ^m  (/) )  <->  f  e.  {
(/) } ) )
98eqrdv 2098 1  |-  ( B  e.  V  ->  ( B  ^m  (/) )  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1299    e. wcel 1448   _Vcvv 2641   (/)c0 3310   {csn 3474   -->wf 5055  (class class class)co 5706    ^m cmap 6472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-map 6474
This theorem is referenced by: (None)
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