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Theorem fndmin 6004
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 5924 . . . . . . 7  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21biimpi 199 . . . . . 6  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 df-mpt 4456 . . . . . 6  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
42, 3syl6eq 2521 . . . . 5  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
5 dffn5 5924 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
65biimpi 199 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
7 df-mpt 4456 . . . . . 6  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
86, 7syl6eq 2521 . . . . 5  |-  ( G  Fn  A  ->  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) } )
94, 8ineqan12d 3627 . . . 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 4970 . . . 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 2521 . . 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 5042 . 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 1842 . . . . 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 844 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <->  ( (
x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) )
1514exbii 1726 . . . . 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 5889 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 eqeq1 2475 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  =  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) ) )
1816, 17ceqsexv 3070 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) )
1918anbi2i 708 . . . . 5  |-  ( ( x  e.  A  /\  E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) ) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2013, 15, 193bitr3i 283 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2120abbii 2587 . . 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 5051 . . 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 2765 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) }
2421, 22, 233eqtr4i 2503 . 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 2521 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   {crab 2760    i^i cin 3389   {copab 4453    |-> cmpt 4454   dom cdm 4839    Fn wfn 5584   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  fneqeql  6005  fninfp  6107  mhmeql  16689  ghmeql  16983  lmhmeql  18356  hauseqlcld  20738  cvmliftmolem1  30076  cvmliftmolem2  30077  hausgraph  36160  mgmhmeql  40311
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