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Theorem f0rn0 5189
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 5152 . . 3  |-  ( E : X --> Y  ->  dom  E  =  X )
2 frn 5155 . . . . . . . . 9  |-  ( E : X --> Y  ->  ran  E  C_  Y )
3 ralnex 2369 . . . . . . . . . 10  |-  ( A. y  e.  Y  -.  y  e.  ran  E  <->  -.  E. y  e.  Y  y  e.  ran  E )
4 disj 3328 . . . . . . . . . . 11  |-  ( ( Y  i^i  ran  E
)  =  (/)  <->  A. y  e.  Y  -.  y  e.  ran  E )
5 df-ss 3010 . . . . . . . . . . . 12  |-  ( ran 
E  C_  Y  <->  ( ran  E  i^i  Y )  =  ran  E )
6 incom 3190 . . . . . . . . . . . . . 14  |-  ( ran 
E  i^i  Y )  =  ( Y  i^i  ran 
E )
76eqeq1i 2095 . . . . . . . . . . . . 13  |-  ( ( ran  E  i^i  Y
)  =  ran  E  <->  ( Y  i^i  ran  E
)  =  ran  E
)
8 eqtr2 2106 . . . . . . . . . . . . . 14  |-  ( ( ( Y  i^i  ran  E )  =  ran  E  /\  ( Y  i^i  ran  E )  =  (/) )  ->  ran  E  =  (/) )
98ex 113 . . . . . . . . . . . . 13  |-  ( ( Y  i^i  ran  E
)  =  ran  E  ->  ( ( Y  i^i  ran 
E )  =  (/)  ->  ran  E  =  (/) ) )
107, 9sylbi 119 . . . . . . . . . . . 12  |-  ( ( ran  E  i^i  Y
)  =  ran  E  ->  ( ( Y  i^i  ran 
E )  =  (/)  ->  ran  E  =  (/) ) )
115, 10sylbi 119 . . . . . . . . . . 11  |-  ( ran 
E  C_  Y  ->  ( ( Y  i^i  ran  E )  =  (/)  ->  ran  E  =  (/) ) )
124, 11syl5bir 151 . . . . . . . . . 10  |-  ( ran 
E  C_  Y  ->  ( A. y  e.  Y  -.  y  e.  ran  E  ->  ran  E  =  (/) ) )
133, 12syl5bir 151 . . . . . . . . 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 122 . . . . . . 7  |-  ( ( E : X --> Y  /\  -.  E. y  e.  Y  y  e.  ran  E )  ->  ran  E  =  (/) )
1615adantl 271 . . . . . 6  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  ran  E  =  (/) )
17 dm0rn0 4641 . . . . . 6  |-  ( dom 
E  =  (/)  <->  ran  E  =  (/) )
1816, 17sylibr 132 . . . . 5  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  dom  E  =  (/) )
19 eqeq1 2094 . . . . . . 7  |-  ( X  =  dom  E  -> 
( X  =  (/)  <->  dom  E  =  (/) ) )
2019eqcoms 2091 . . . . . 6  |-  ( dom 
E  =  X  -> 
( X  =  (/)  <->  dom  E  =  (/) ) )
2120adantr 270 . . . . 5  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  ( X  =  (/) 
<->  dom  E  =  (/) ) )
2218, 21mpbird 165 . . . 4  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  X  =  (/) )
2322exp32 357 . . 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 122 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360    i^i cin 2996    C_ wss 2997   (/)c0 3284   dom cdm 4428   ran crn 4429   -->wf 4998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436  df-dm 4438  df-rn 4439  df-fn 5005  df-f 5006
This theorem is referenced by: (None)
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