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Theorem funfveu 5509
Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
funfveu  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F
y )
Distinct variable groups:    y, A    y, F

Proof of Theorem funfveu
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2233 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 461 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 breq1 3992 . . . . 5  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
43eubidv 2027 . . . 4  |-  ( x  =  A  ->  ( E! y  x F
y  <->  E! y  A F y ) )
52, 4imbi12d 233 . . 3  |-  ( x  =  A  ->  (
( ( Fun  F  /\  x  e.  dom  F )  ->  E! y  x F y )  <->  ( ( Fun  F  /\  A  e. 
dom  F )  ->  E! y  A F
y ) ) )
6 dffun8 5226 . . . . 5  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x  e.  dom  F E! y  x F y ) )
76simprbi 273 . . . 4  |-  ( Fun 
F  ->  A. x  e.  dom  F E! y  x F y )
87r19.21bi 2558 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E! y  x F
y )
95, 8vtoclg 2790 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F y ) )
109anabsi7 576 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F
y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348   E!weu 2019    e. wcel 2141   A.wral 2448   class class class wbr 3989   dom cdm 4611   Rel wrel 4616   Fun wfun 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-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 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  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-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200
This theorem is referenced by:  funfvex  5513
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