MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fndmdif Structured version   Unicode version

Theorem fndmdif 5863
Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdif  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
Distinct variable groups:    x, F    x, G    x, A

Proof of Theorem fndmdif
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 difss 3460 . . . . 5  |-  ( F 
\  G )  C_  F
2 dmss 5098 . . . . 5  |-  ( ( F  \  G ) 
C_  F  ->  dom  ( F  \  G ) 
C_  dom  F )
31, 2ax-mp 5 . . . 4  |-  dom  ( F  \  G )  C_  dom  F
4 fndm 5573 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
54adantr 453 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
63, 5syl5sseq 3382 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  C_  A
)
7 dfss1 3531 . . 3  |-  ( dom  ( F  \  G
)  C_  A  <->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
86, 7sylib 190 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
9 vex 2965 . . . . 5  |-  x  e. 
_V
109eldm 5096 . . . 4  |-  ( x  e.  dom  ( F 
\  G )  <->  E. y  x ( F  \  G ) y )
11 eqcom 2444 . . . . . . . . 9  |-  ( ( F `  x )  =  ( G `  x )  <->  ( G `  x )  =  ( F `  x ) )
12 fnbrfvb 5796 . . . . . . . . 9  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( G `  x )  =  ( F `  x )  <-> 
x G ( F `
 x ) ) )
1311, 12syl5bb 250 . . . . . . . 8  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( G `  x )  <-> 
x G ( F `
 x ) ) )
1413adantll 696 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =  ( G `
 x )  <->  x G
( F `  x
) ) )
1514necon3abid 2640 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  -.  x G ( F `  x ) ) )
16 fvex 5771 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 breq2 4241 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
x G y  <->  x G
( F `  x
) ) )
1817notbid 287 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( -.  x G y  <->  -.  x G ( F `  x ) ) )
1916, 18ceqsexv 2997 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  -.  x G ( F `  x ) )
2015, 19syl6bbr 256 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  E. y
( y  =  ( F `  x )  /\  -.  x G y ) ) )
21 eqcom 2444 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
22 fnbrfvb 5796 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2321, 22syl5bb 250 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
2423adantlr 697 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
2524anbi1d 687 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  ( x F y  /\  -.  x G y ) ) )
26 brdif 4285 . . . . . . 7  |-  ( x ( F  \  G
) y  <->  ( x F y  /\  -.  x G y ) )
2725, 26syl6bbr 256 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  x ( F  \  G ) y ) )
2827exbidv 1637 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  E. y  x ( F  \  G ) y ) )
2920, 28bitr2d 247 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y  x ( F  \  G ) y  <-> 
( F `  x
)  =/=  ( G `
 x ) ) )
3010, 29syl5bb 250 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
x  e.  dom  ( F  \  G )  <->  ( F `  x )  =/=  ( G `  x )
) )
3130rabbi2dva 3534 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) } )
328, 31eqtr3d 2476 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1727    =/= wne 2605   {crab 2715    \ cdif 3303    i^i cin 3305    C_ wss 3306   class class class wbr 4237   dom cdm 4907    Fn wfn 5478   ` cfv 5483
This theorem is referenced by:  fndmdifcom  5864  fndmdifeq0  5865  wemapso2lem  7548  wemapso2  7550  ptcmplem2  18115  fndifnfp  26775  dsmmbas2  27218  frlmbas  27238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fn 5486  df-fv 5491
  Copyright terms: Public domain W3C validator