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Theorem fvreseq 5792
Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fvreseq  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Distinct variable groups:    x, B    x, F    x, G
Allowed substitution hint:    A( x)

Proof of Theorem fvreseq
StepHypRef Expression
1 fnssres 5517 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fnssres 5517 . . . 4  |-  ( ( G  Fn  A  /\  B  C_  A )  -> 
( G  |`  B )  Fn  B )
31, 2anim12i 550 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  ( G  Fn  A  /\  B  C_  A
) )  ->  (
( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B ) )
43anandirs 805 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B
) )
5 eqfnfv 5786 . . 3  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( ( F  |`  B ) `  x )  =  ( ( G  |`  B ) `
 x ) ) )
6 fvres 5704 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
7 fvres 5704 . . . . 5  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
86, 7eqeq12d 2418 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  =  ( ( G  |`  B ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
98ralbiia 2698 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
)  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
105, 9syl6bb 253 . 2  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
114, 10syl 16 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280    |` cres 4839    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  tfrlem1  6595  tfr3  6619  fseqenlem1  7861  dchrresb  20996  rdgprc  25365  predreseq  25393  wfr3g  25469  frr3g  25494  bnj1536  28931  bnj1253  29092  bnj1280  29095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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