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Theorem fndmin 5406
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 dffn5im 5350 . . . . . 6  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
2 df-mpt 3901 . . . . . 6  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
31, 2syl6eq 2136 . . . . 5  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
4 dffn5im 5350 . . . . . 6  |-  ( G  Fn  A  ->  G  =  ( x  e.  A  |->  ( G `  x ) ) )
5 df-mpt 3901 . . . . . 6  |-  ( x  e.  A  |->  ( G `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) }
64, 5syl6eq 2136 . . . . 5  |-  ( G  Fn  A  ->  G  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( G `  x ) ) } )
73, 6ineqan12d 3203 . . . 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
) ) } ) )
8 inopab 4568 . . . 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 ) ) ) }
97, 8syl6eq 2136 . . 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 ) ) ) } )
109dmeqd 4638 . 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 )
) ) } )
11 anandi 557 . . . . . . . 8  |-  ( ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <->  ( (
x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) )
1211exbii 1541 . . . . . . 7  |-  ( 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 )
) ) )
13 19.42v 1834 . . . . . . 7  |-  ( E. y ( x  e.  A  /\  ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <-> 
( x  e.  A  /\  E. y ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) ) )
1412, 13bitr3i 184 . . . . . 6  |-  ( E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) )  <->  ( x  e.  A  /\  E. y
( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) ) )
15 funfvex 5322 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
16 eqeq1 2094 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
y  =  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) ) )
1716ceqsexgv 2746 . . . . . . . . 9  |-  ( ( F `  x )  e.  _V  ->  ( E. y ( y  =  ( F `  x
)  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) ) )
1815, 17syl 14 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( E. y ( y  =  ( F `
 x )  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) ) )
1918funfni 5114 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( E. y ( y  =  ( F `
 x )  /\  y  =  ( G `  x ) )  <->  ( F `  x )  =  ( G `  x ) ) )
2019pm5.32da 440 . . . . . 6  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  E. y ( y  =  ( F `  x )  /\  y  =  ( G `  x ) ) )  <-> 
( x  e.  A  /\  ( F `  x
)  =  ( G `
 x ) ) ) )
2114, 20syl5bb 190 . . . . 5  |-  ( F  Fn  A  ->  ( E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) )  <->  ( x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) ) )
2221abbidv 2205 . . . 4  |-  ( F  Fn  A  ->  { x  |  E. y ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) } )
23 dmopab 4647 . . . 4  |-  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 )
) ) }
24 df-rab 2368 . . . 4  |-  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  { x  |  (
x  e.  A  /\  ( F `  x )  =  ( G `  x ) ) }
2522, 23, 243eqtr4g 2145 . . 3  |-  ( F  Fn  A  ->  dom  {
<. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x ) )  /\  ( x  e.  A  /\  y  =  ( G `  x )
) ) }  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
2625adantr 270 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  =  ( F `  x )
)  /\  ( x  e.  A  /\  y  =  ( G `  x ) ) ) }  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
2710, 26eqtrd 2120 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 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   {crab 2363   _Vcvv 2619    i^i cin 2998   {copab 3898    |-> cmpt 3899   dom cdm 4438   Fun wfun 5009    Fn wfn 5010   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  fneqeql  5407
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