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Theorem fun2ssres 5295
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 4984 . . . 4  |-  ( A 
C_  dom  G  ->  ( ( F  |`  dom  G
)  |`  A )  =  ( F  |`  A ) )
21eqcomd 2288 . . 3  |-  ( A 
C_  dom  G  ->  ( F  |`  A )  =  ( ( F  |`  dom  G )  |`  A ) )
3 funssres 5294 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43reseq1d 4954 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
)  |`  A )  =  ( G  |`  A ) )
52, 4sylan9eqr 2337 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A )
)
653impa 1146 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 358    /\ w3a 934    = wceq 1623    C_ wss 3152   dom cdm 4689    |` cres 4691   Fun wfun 5249
This theorem is referenced by:  tfrlem9  6401  tfrlem9a  6402  tfrlem11  6404  subgores  20971  wfrlem12  24267  wfrlem14  24269  frrlem11  24293  bnj1503  28881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-fun 5257
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