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Theorem fun2ssres 5494
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 5175 . . . 4  |-  ( A 
C_  dom  G  ->  ( ( F  |`  dom  G
)  |`  A )  =  ( F  |`  A ) )
21eqcomd 2441 . . 3  |-  ( A 
C_  dom  G  ->  ( F  |`  A )  =  ( ( F  |`  dom  G )  |`  A ) )
3 funssres 5493 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43reseq1d 5145 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
)  |`  A )  =  ( G  |`  A ) )
52, 4sylan9eqr 2490 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  C_  dom  G )  ->  ( F  |`  A )  =  ( G  |`  A )
)
653impa 1148 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 359    /\ w3a 936    = wceq 1652    C_ wss 3320   dom cdm 4878    |` cres 4880   Fun wfun 5448
This theorem is referenced by:  tfrlem9  6646  tfrlem9a  6647  tfrlem11  6649  subgores  21892  wfrlem12  25549  wfrlem14  25551  frrlem11  25594  bnj1503  29220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-fun 5456
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