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Theorem f0rn0 5429
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 5390 . . 3  |-  ( E : X --> Y  ->  dom  E  =  X )
2 frn 5393 . . . . . . . . 9  |-  ( E : X --> Y  ->  ran  E  C_  Y )
3 ralnex 2478 . . . . . . . . . 10  |-  ( A. y  e.  Y  -.  y  e.  ran  E  <->  -.  E. y  e.  Y  y  e.  ran  E )
4 disj 3486 . . . . . . . . . . 11  |-  ( ( Y  i^i  ran  E
)  =  (/)  <->  A. y  e.  Y  -.  y  e.  ran  E )
5 df-ss 3157 . . . . . . . . . . . 12  |-  ( ran 
E  C_  Y  <->  ( ran  E  i^i  Y )  =  ran  E )
6 incom 3342 . . . . . . . . . . . . . 14  |-  ( ran 
E  i^i  Y )  =  ( Y  i^i  ran 
E )
76eqeq1i 2197 . . . . . . . . . . . . 13  |-  ( ( ran  E  i^i  Y
)  =  ran  E  <->  ( Y  i^i  ran  E
)  =  ran  E
)
8 eqtr2 2208 . . . . . . . . . . . . . 14  |-  ( ( ( Y  i^i  ran  E )  =  ran  E  /\  ( Y  i^i  ran  E )  =  (/) )  ->  ran  E  =  (/) )
98ex 115 . . . . . . . . . . . . 13  |-  ( ( Y  i^i  ran  E
)  =  ran  E  ->  ( ( Y  i^i  ran 
E )  =  (/)  ->  ran  E  =  (/) ) )
107, 9sylbi 121 . . . . . . . . . . . 12  |-  ( ( ran  E  i^i  Y
)  =  ran  E  ->  ( ( Y  i^i  ran 
E )  =  (/)  ->  ran  E  =  (/) ) )
115, 10sylbi 121 . . . . . . . . . . 11  |-  ( ran 
E  C_  Y  ->  ( ( Y  i^i  ran  E )  =  (/)  ->  ran  E  =  (/) ) )
124, 11biimtrrid 153 . . . . . . . . . 10  |-  ( ran 
E  C_  Y  ->  ( A. y  e.  Y  -.  y  e.  ran  E  ->  ran  E  =  (/) ) )
133, 12biimtrrid 153 . . . . . . . . 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 124 . . . . . . 7  |-  ( ( E : X --> Y  /\  -.  E. y  e.  Y  y  e.  ran  E )  ->  ran  E  =  (/) )
1615adantl 277 . . . . . 6  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  ran  E  =  (/) )
17 dm0rn0 4862 . . . . . 6  |-  ( dom 
E  =  (/)  <->  ran  E  =  (/) )
1816, 17sylibr 134 . . . . 5  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  dom  E  =  (/) )
19 eqeq1 2196 . . . . . . 7  |-  ( X  =  dom  E  -> 
( X  =  (/)  <->  dom  E  =  (/) ) )
2019eqcoms 2192 . . . . . 6  |-  ( dom 
E  =  X  -> 
( X  =  (/)  <->  dom  E  =  (/) ) )
2120adantr 276 . . . . 5  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  ( X  =  (/) 
<->  dom  E  =  (/) ) )
2218, 21mpbird 167 . . . 4  |-  ( ( dom  E  =  X  /\  ( E : X
--> Y  /\  -.  E. y  e.  Y  y  e.  ran  E ) )  ->  X  =  (/) )
2322exp32 365 . . 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 124 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 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469    i^i cin 3143    C_ wss 3144   (/)c0 3437   dom cdm 4644   ran crn 4645   -->wf 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4652  df-dm 4654  df-rn 4655  df-fn 5238  df-f 5239
This theorem is referenced by: (None)
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