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Theorem fvreseq 5492
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 5206 . . . 4  |-  ( ( F  Fn  A  /\  B  C_  A )  -> 
( F  |`  B )  Fn  B )
2 fnssres 5206 . . . 4  |-  ( ( G  Fn  A  /\  B  C_  A )  -> 
( G  |`  B )  Fn  B )
31, 2anim12i 336 . . 3  |-  ( ( ( F  Fn  A  /\  B  C_  A )  /\  ( G  Fn  A  /\  B  C_  A
) )  ->  (
( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B ) )
43anandirs 567 . 2  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  B  C_  A
)  ->  ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B
) )
5 eqfnfv 5486 . . 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 5413 . . . . 5  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
7 fvres 5413 . . . . 5  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
86, 7eqeq12d 2132 . . . 4  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  =  ( ( G  |`  B ) `  x )  <->  ( F `  x )  =  ( G `  x ) ) )
98ralbiia 2426 . . 3  |-  ( A. x  e.  B  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
)  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
105, 9syl6bb 195 . 2  |-  ( ( ( F  |`  B )  Fn  B  /\  ( G  |`  B )  Fn  B )  ->  (
( F  |`  B )  =  ( G  |`  B )  <->  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) )
114, 10syl 14 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    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393    C_ wss 3041    |` cres 4511    Fn wfn 5088   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  tfri3  6232
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