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Theorem fvresd 5579
Description: The value of a restricted function, deduction version of fvres 5578. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
fvresd.1  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
fvresd  |-  ( ph  ->  ( ( F  |`  B ) `  A
)  =  ( F `
 A ) )

Proof of Theorem fvresd
StepHypRef Expression
1 fvresd.1 . 2  |-  ( ph  ->  A  e.  B )
2 fvres 5578 . 2  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
31, 2syl 14 1  |-  ( ph  ->  ( ( F  |`  B ) `  A
)  =  ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2164    |` cres 4661   ` cfv 5254
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-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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-res 4671  df-iota 5215  df-fv 5262
This theorem is referenced by:  difinfsn  7159  seqf1oglem2  10591  gsumsplit1r  12981  resmhm  13059  resghm  13330  upxp  14440  uptx  14442  reeflog  14998  relogef  14999  trilpolemlt1  15531
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