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Theorem fun2ssres 5311
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 5000 . . . 4  |-  ( A 
C_  dom  G  ->  ( ( F  |`  dom  G
)  |`  A )  =  ( F  |`  A ) )
21eqcomd 2301 . . 3  |-  ( A 
C_  dom  G  ->  ( F  |`  A )  =  ( ( F  |`  dom  G )  |`  A ) )
3 funssres 5310 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43reseq1d 4970 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
)  |`  A )  =  ( G  |`  A ) )
52, 4sylan9eqr 2350 . 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 1632    C_ wss 3165   dom cdm 4705    |` cres 4707   Fun wfun 5265
This theorem is referenced by:  tfrlem9  6417  tfrlem9a  6418  tfrlem11  6420  subgores  20987  wfrlem12  24338  wfrlem14  24340  frrlem11  24364  bnj1503  29197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-fun 5273
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