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Theorem fvresd 5552
Description: The value of a restricted function, deduction version of fvres 5551. (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 5551 . 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 1363    e. wcel 2158    |` cres 4640   ` cfv 5228
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-res 4650  df-iota 5190  df-fv 5236
This theorem is referenced by:  difinfsn  7112  upxp  13999  uptx  14001  reeflog  14511  relogef  14512  trilpolemlt1  15017
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