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Theorem fndmin 5970
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 5893 . . . . . . 7  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21biimpi 194 . . . . . 6  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 df-mpt 4499 . . . . . 6  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
42, 3syl6eq 2511 . . . . 5  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
5 dffn5 5893 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
65biimpi 194 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
7 df-mpt 4499 . . . . . 6  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
86, 7syl6eq 2511 . . . . 5  |-  ( G  Fn  A  ->  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) } )
94, 8ineqan12d 3688 . . . 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 5122 . . . 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 2511 . . 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 5194 . 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 1780 . . . . 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 826 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <->  ( (
x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) )
1514exbii 1672 . . . . 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 5858 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 eqeq1 2458 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  =  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) ) )
1816, 17ceqsexv 3143 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) )
1918anbi2i 692 . . . . 5  |-  ( ( x  e.  A  /\  E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) ) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2013, 15, 193bitr3i 275 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2120abbii 2588 . . 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 5202 . . 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 2813 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) }
2421, 22, 233eqtr4i 2493 . 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 2511 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 367    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   {crab 2808    i^i cin 3460   {copab 4496    |-> cmpt 4497   dom cdm 4988    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  fneqeql  5971  fninfp  6074  mhmeql  16194  ghmeql  16488  lmhmeql  17896  hauseqlcld  20313  cvmliftmolem1  28990  cvmliftmolem2  28991  hausgraph  31413  mgmhmeql  32863
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