Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqfnun Unicode version

Theorem eqfnun 26490
Description: Two functions on  A  u.  B are equal if and only if they have equal restrictions to both  A and  B. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
eqfnun  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( F  =  G  <-> 
( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )

Proof of Theorem eqfnun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 reseq1 4965 . . 3  |-  ( F  =  G  ->  ( F  |`  A )  =  ( G  |`  A ) )
2 reseq1 4965 . . 3  |-  ( F  =  G  ->  ( F  |`  B )  =  ( G  |`  B ) )
31, 2jca 518 . 2  |-  ( F  =  G  ->  (
( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) )
4 elun 3329 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
5 fveq1 5540 . . . . . . . . 9  |-  ( ( F  |`  A )  =  ( G  |`  A )  ->  (
( F  |`  A ) `
 x )  =  ( ( G  |`  A ) `  x
) )
6 fvres 5558 . . . . . . . . 9  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
75, 6sylan9req 2349 . . . . . . . 8  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  (
( G  |`  A ) `
 x )  =  ( F `  x
) )
8 fvres 5558 . . . . . . . . 9  |-  ( x  e.  A  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
98adantl 452 . . . . . . . 8  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
107, 9eqtr3d 2330 . . . . . . 7  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
1110adantlr 695 . . . . . 6  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  A )  ->  ( F `  x
)  =  ( G `
 x ) )
12 fveq1 5540 . . . . . . . . 9  |-  ( ( F  |`  B )  =  ( G  |`  B )  ->  (
( F  |`  B ) `
 x )  =  ( ( G  |`  B ) `  x
) )
13 fvres 5558 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
1412, 13sylan9req 2349 . . . . . . . 8  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  (
( G  |`  B ) `
 x )  =  ( F `  x
) )
15 fvres 5558 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
1615adantl 452 . . . . . . . 8  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  (
( G  |`  B ) `
 x )  =  ( G `  x
) )
1714, 16eqtr3d 2330 . . . . . . 7  |-  ( ( ( F  |`  B )  =  ( G  |`  B )  /\  x  e.  B )  ->  ( F `  x )  =  ( G `  x ) )
1817adantll 694 . . . . . 6  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  B )  ->  ( F `  x
)  =  ( G `
 x ) )
1911, 18jaodan 760 . . . . 5  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  ( x  e.  A  \/  x  e.  B
) )  ->  ( F `  x )  =  ( G `  x ) )
204, 19sylan2b 461 . . . 4  |-  ( ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  /\  x  e.  ( A  u.  B ) )  -> 
( F `  x
)  =  ( G `
 x ) )
2120ralrimiva 2639 . . 3  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  ->  A. x  e.  ( A  u.  B
) ( F `  x )  =  ( G `  x ) )
22 eqfnfv 5638 . . 3  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( F  =  G  <->  A. x  e.  ( A  u.  B )
( F `  x
)  =  ( G `
 x ) ) )
2321, 22syl5ibr 212 . 2  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( ( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) )  ->  F  =  G )
)
243, 23impbid2 195 1  |-  ( ( F  Fn  ( A  u.  B )  /\  G  Fn  ( A  u.  B ) )  -> 
( F  =  G  <-> 
( ( F  |`  A )  =  ( G  |`  A )  /\  ( F  |`  B )  =  ( G  |`  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163    |` cres 4707    Fn wfn 5266   ` cfv 5271
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-13 1698  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-pow 4204  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-sbc 3005  df-csb 3095  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-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
  Copyright terms: Public domain W3C validator