ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f0rn0 Unicode version

Theorem f0rn0 5390
Description: If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
Assertion
Ref Expression
f0rn0  |-  ( ( E : X --> Y  /\  -.  E. y  e.  Y  y  e.  ran  E )  ->  X  =  (/) )
Distinct variable groups:    y, E    y, Y
Allowed substitution hint:    X( y)

Proof of Theorem f0rn0
StepHypRef Expression
1 fdm 5351 . . 3  |-  ( E : X --> Y  ->  dom  E  =  X )
2 frn 5354 . . . . . . . . 9  |-  ( E : X --> Y  ->  ran  E  C_  Y )
3 ralnex 2458 . . . . . . . . . 10  |-  ( A. y  e.  Y  -.  y  e.  ran  E  <->  -.  E. y  e.  Y  y  e.  ran  E )
4 disj 3462 . . . . . . . . . . 11  |-  ( ( Y  i^i  ran  E
)  =  (/)  <->  A. y  e.  Y  -.  y  e.  ran  E )
5 df-ss 3134 . . . . . . . . . . . 12  |-  ( ran 
E  C_  Y  <->  ( ran  E  i^i  Y )  =  ran  E )
6 incom 3319 . . . . . . . . . . . . . 14  |-  ( ran 
E  i^i  Y )  =  ( Y  i^i  ran 
E )
76eqeq1i 2178 . . . . . . . . . . . . 13  |-  ( ( ran  E  i^i  Y
)  =  ran  E  <->  ( Y  i^i  ran  E
)  =  ran  E
)
8 eqtr2 2189 . . . . . . . . . . . . . 14  |-  ( ( ( Y  i^i  ran  E )  =  ran  E  /\  ( Y  i^i  ran  E )  =  (/) )  ->  ran  E  =  (/) )
98ex 114 . . . . . . . . . . . . 13  |-  ( ( Y  i^i  ran  E
)  =  ran  E  ->  ( ( Y  i^i  ran 
E )  =  (/)  ->  ran  E  =  (/) ) )
107, 9sylbi 120 . . . . . . . . . . . 12  |-  ( ( ran  E  i^i  Y
)  =  ran  E  ->  ( ( Y  i^i  ran 
E )  =  (/)  ->  ran  E  =  (/) ) )
115, 10sylbi 120 . . . . . . . . . . 11  |-  ( ran 
E  C_  Y  ->  ( ( Y  i^i  ran  E )  =  (/)  ->  ran  E  =  (/) ) )
124, 11syl5bir 152 . . . . . . . . . 10  |-  ( ran 
E  C_  Y  ->  ( A. y  e.  Y  -.  y  e.  ran  E  ->  ran  E  =  (/) ) )
133, 12syl5bir 152 . . . . . . . . 9  |-  ( ran 
E  C_  Y  ->  ( -.  E. y  e.  Y  y  e.  ran  E  ->  ran  E  =  (/) ) )
142, 13syl 14 . . . . . . . 8  |-  ( E : X --> Y  -> 
( -.  E. y  e.  Y  y  e.  ran  E  ->  ran  E  =  (/) ) )
1514imp 123 . . . . . . 7  |-  ( ( E : X --> Y  /\  -.  E. y  e.  Y  y  e.  ran  E )  ->  ran  E  =  (/) )
1615adantl 275 . . . . . 6  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  ran  E  =  (/) )
17 dm0rn0 4826 . . . . . 6  |-  ( dom 
E  =  (/)  <->  ran  E  =  (/) )
1816, 17sylibr 133 . . . . 5  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  dom  E  =  (/) )
19 eqeq1 2177 . . . . . . 7  |-  ( X  =  dom  E  -> 
( X  =  (/)  <->  dom  E  =  (/) ) )
2019eqcoms 2173 . . . . . 6  |-  ( dom 
E  =  X  -> 
( X  =  (/)  <->  dom  E  =  (/) ) )
2120adantr 274 . . . . 5  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  ( X  =  (/) 
<->  dom  E  =  (/) ) )
2218, 21mpbird 166 . . . 4  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  X  =  (/) )
2322exp32 363 . . 3  |-  ( dom 
E  =  X  -> 
( E : X --> Y  ->  ( -.  E. y  e.  Y  y  e.  ran  E  ->  X  =  (/) ) ) )
241, 23mpcom 36 . 2  |-  ( E : X --> Y  -> 
( -.  E. y  e.  Y  y  e.  ran  E  ->  X  =  (/) ) )
2524imp 123 1  |-  ( ( E : X --> Y  /\  -.  E. y  e.  Y  y  e.  ran  E )  ->  X  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449    i^i cin 3120    C_ wss 3121   (/)c0 3414   dom cdm 4609   ran crn 4610   -->wf 5192
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-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
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-ral 2453  df-rex 2454  df-v 2732  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-br 3988  df-opab 4049  df-cnv 4617  df-dm 4619  df-rn 4620  df-fn 5199  df-f 5200
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator