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Theorem mapdm0 6637
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 4114 . . . . 5  |-  (/)  e.  _V
2 elmapg 6635 . . . . 5  |-  ( ( B  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( B  ^m  (/) )  <->  f : (/) --> B ) )
31, 2mpan2 423 . . . 4  |-  ( B  e.  V  ->  (
f  e.  ( B  ^m  (/) )  <->  f : (/) --> B ) )
4 f0bi 5388 . . . 4  |-  ( f : (/) --> B  <->  f  =  (/) )
53, 4bitrdi 195 . . 3  |-  ( B  e.  V  ->  (
f  e.  ( B  ^m  (/) )  <->  f  =  (/) ) )
6 vex 2733 . . . 4  |-  f  e. 
_V
76elsn 3597 . . 3  |-  ( f  e.  { (/) }  <->  f  =  (/) )
85, 7bitr4di 197 . 2  |-  ( B  e.  V  ->  (
f  e.  ( B  ^m  (/) )  <->  f  e.  {
(/) } ) )
98eqrdv 2168 1  |-  ( B  e.  V  ->  ( B  ^m  (/) )  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   (/)c0 3414   {csn 3581   -->wf 5192  (class class class)co 5850    ^m cmap 6622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-map 6624
This theorem is referenced by: (None)
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