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Theorem funresdfunsnss 5589
Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
Assertion
Ref Expression
funresdfunsnss  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  C_  F
)

Proof of Theorem funresdfunsnss
StepHypRef Expression
1 funrel 5108 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
2 resdmdfsn 4830 . . . . 5  |-  ( Rel 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
31, 2syl 14 . . . 4  |-  ( Fun 
F  ->  ( F  |`  ( _V  \  { X } ) )  =  ( F  |`  ( dom  F  \  { X } ) ) )
43adantr 272 . . 3  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( F  |`  ( _V  \  { X }
) )  =  ( F  |`  ( dom  F 
\  { X }
) ) )
54uneq1d 3197 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  =  ( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
6 funfn 5121 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
7 fnsnsplitss 5585 . . 3  |-  ( ( F  Fn  dom  F  /\  X  e.  dom  F )  ->  ( ( F  |`  ( dom  F  \  { X } ) )  u.  { <. X ,  ( F `  X ) >. } ) 
C_  F )
86, 7sylanb 280 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( dom  F  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  C_  F
)
95, 8eqsstrd 3101 1  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } )  C_  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   _Vcvv 2658    \ cdif 3036    u. cun 3037    C_ wss 3039   {csn 3495   <.cop 3498   dom cdm 4507    |` cres 4509   Rel wrel 4512   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099
This theorem is referenced by: (None)
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