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Theorem fndmin 5648
Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmin  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
Distinct variable groups:    x, F    x, G    x, A

Proof of Theorem fndmin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dffn5 5584 . . . . . . 7  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21biimpi 186 . . . . . 6  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 df-mpt 4095 . . . . . 6  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
42, 3syl6eq 2344 . . . . 5  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
5 dffn5 5584 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
65biimpi 186 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
7 df-mpt 4095 . . . . . 6  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
86, 7syl6eq 2344 . . . . 5  |-  ( G  Fn  A  ->  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) } )
94, 8ineqan12d 3385 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  i^i  G
)  =  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x
) ) }  i^i  {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x
) ) } ) )
10 inopab 4832 . . . 4  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x
) ) }  i^i  {
<. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x
) ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x )
)  /\  ( x  e.  A  /\  y  =  ( G `  x ) ) ) }
119, 10syl6eq 2344 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  i^i  G
)  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x )
)  /\  ( x  e.  A  /\  y  =  ( G `  x ) ) ) } )
1211dmeqd 4897 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  dom  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) } )
13 19.42v 1858 . . . . 5  |-  ( E. y ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <-> 
( x  e.  A  /\  E. y ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) ) )
14 anandi 801 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <->  ( (
x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) )
1514exbii 1572 . . . . 5  |-  ( E. y ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <->  E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) )
16 fvex 5555 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 eqeq1 2302 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  =  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) ) )
1816, 17ceqsexv 2836 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) )
1918anbi2i 675 . . . . 5  |-  ( ( x  e.  A  /\  E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) ) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2013, 15, 193bitr3i 266 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2120abbii 2408 . . 3  |-  { x  |  E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) }
22 dmopab 4905 . . 3  |-  dom  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) }  =  { x  |  E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) }
23 df-rab 2565 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) }
2421, 22, 233eqtr4i 2326 . 2  |-  dom  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) }  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }
2512, 24syl6eq 2344 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    i^i cin 3164   {copab 4092    e. cmpt 4093   dom cdm 4705    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  fneqeql  5649  mhmeql  14457  ghmeql  14721  lmhmeql  15828  hauseqlcld  17356  cvmliftmolem1  23827  cvmliftmolem2  23828  itgaddnclem2  25010  fninfp  26857  hausgraph  27634
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  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-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-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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