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Theorem funfveu 5483
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 2220 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 460 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 breq1 3970 . . . . 5  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
43eubidv 2014 . . . 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 5200 . . . . 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 2545 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E! y  x F
y )
95, 8vtoclg 2772 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F y ) )
109anabsi7 571 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 1335   E!weu 2006    e. wcel 2128   A.wral 2435   class class class wbr 3967   dom cdm 4588   Rel wrel 4593   Fun wfun 5166
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-id 4255  df-cnv 4596  df-co 4597  df-dm 4598  df-fun 5174
This theorem is referenced by:  funfvex  5487
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