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Theorem fun2ssres 5629
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 5302 . . . 4  |-  ( A 
C_  dom  G  ->  ( ( F  |`  dom  G
)  |`  A )  =  ( F  |`  A ) )
21eqcomd 2475 . . 3  |-  ( A 
C_  dom  G  ->  ( F  |`  A )  =  ( ( F  |`  dom  G )  |`  A ) )
3 funssres 5628 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43reseq1d 5272 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
)  |`  A )  =  ( G  |`  A ) )
52, 4sylan9eqr 2530 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A )
)
653impa 1191 1  |-  ( ( Fun  F  /\  G  C_  F  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    C_ wss 3476   dom cdm 4999    |` cres 5001   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-fun 5590
This theorem is referenced by:  tfrlem9  7055  tfrlem9a  7056  tfrlem11  7058  subgores  25079  wfrlem12  29207  wfrlem14  29209  frrlem11  29252  bnj1503  33203
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