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Theorem fun2ssres 5314
Description: Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
fun2ssres  |-  ( ( Fun  F  /\  G  C_  F  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A ) )

Proof of Theorem fun2ssres
StepHypRef Expression
1 resabs1 4988 . . . 4  |-  ( A 
C_  dom  G  ->  ( ( F  |`  dom  G
)  |`  A )  =  ( F  |`  A ) )
21eqcomd 2211 . . 3  |-  ( A 
C_  dom  G  ->  ( F  |`  A )  =  ( ( F  |`  dom  G )  |`  A ) )
3 funssres 5313 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43reseq1d 4958 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
)  |`  A )  =  ( G  |`  A ) )
52, 4sylan9eqr 2260 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A )
)
653impa 1197 1  |-  ( ( Fun  F  /\  G  C_  F  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 981    = wceq 1373    C_ wss 3166   dom cdm 4675    |` cres 4677   Fun wfun 5265
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-res 4687  df-fun 5273
This theorem is referenced by:  tfrlem9  6405  tfrlemiubacc  6416  tfr1onlemubacc  6432  tfrcllemubacc  6445
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