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Theorem fndmin 5838
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 5773 . . . . . . 7  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
21biimpi 188 . . . . . 6  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
3 df-mpt 4269 . . . . . 6  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
42, 3syl6eq 2485 . . . . 5  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
5 dffn5 5773 . . . . . . 7  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
65biimpi 188 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
7 df-mpt 4269 . . . . . 6  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
86, 7syl6eq 2485 . . . . 5  |-  ( G  Fn  A  ->  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) } )
94, 8ineqan12d 3545 . . . 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 5006 . . . 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 2485 . . 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 5073 . 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 1929 . . . . 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 803 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <->  ( (
x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) )
1514exbii 1593 . . . . 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 5743 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 eqeq1 2443 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  =  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) ) )
1816, 17ceqsexv 2992 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) )
1918anbi2i 677 . . . . 5  |-  ( ( x  e.  A  /\  E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) ) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2013, 15, 193bitr3i 268 . . . 4  |-  ( E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) )
2120abbii 2549 . . 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 5081 . . 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 2715 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) }
2421, 22, 233eqtr4i 2467 . 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 2485 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 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2423   {crab 2710    i^i cin 3320   {copab 4266    e. cmpt 4267   dom cdm 4879    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  fneqeql  5839  mhmeql  14765  ghmeql  15029  lmhmeql  16132  hauseqlcld  17679  cvmliftmolem1  24969  cvmliftmolem2  24970  fninfp  26736  hausgraph  27509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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