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Theorem fgraphopab 26940
Description: Express a function as a subset of the cross product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
fgraphopab  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
Distinct variable groups:    F, a,
b    A, a, b    B, a, b

Proof of Theorem fgraphopab
StepHypRef Expression
1 fssxp 5366 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
2 df-ss 3167 . . . 4  |-  ( F 
C_  ( A  X.  B )  <->  ( F  i^i  ( A  X.  B
) )  =  F )
31, 2sylib 188 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  F )
4 ffn 5355 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
5 dffn5 5530 . . . . 5  |-  ( F  Fn  A  <->  F  =  ( a  e.  A  |->  ( F `  a
) ) )
64, 5sylib 188 . . . 4  |-  ( F : A --> B  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
76ineq1d 3370 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  ( ( a  e.  A  |->  ( F `
 a ) )  i^i  ( A  X.  B ) ) )
83, 7eqtr3d 2318 . 2  |-  ( F : A --> B  ->  F  =  ( (
a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) ) )
9 df-mpt 4080 . . . 4  |-  ( a  e.  A  |->  ( F `
 a ) )  =  { <. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a ) ) }
10 df-xp 4694 . . . 4  |-  ( A  X.  B )  =  { <. a ,  b
>.  |  ( a  e.  A  /\  b  e.  B ) }
119, 10ineq12i 3369 . . 3  |-  ( ( a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) )  =  ( { <. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a ) ) }  i^i  { <. a ,  b >.  |  ( a  e.  A  /\  b  e.  B ) } )
12 inopab 4815 . . 3  |-  ( {
<. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a
) ) }  i^i  {
<. a ,  b >.  |  ( a  e.  A  /\  b  e.  B ) } )  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  =  ( F `  a )
)  /\  ( a  e.  A  /\  b  e.  B ) ) }
13 anandi 801 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  =  ( F `  a ) )  /\  ( a  e.  A  /\  b  e.  B
) ) )
14 ancom 437 . . . . . . 7  |-  ( ( b  =  ( F `
 a )  /\  b  e.  B )  <->  ( b  e.  B  /\  b  =  ( F `  a ) ) )
1514anbi2i 675 . . . . . 6  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( a  e.  A  /\  (
b  e.  B  /\  b  =  ( F `  a ) ) ) )
16 anass 630 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( a  e.  A  /\  ( b  e.  B  /\  b  =  ( F `  a ) ) ) )
17 eqcom 2286 . . . . . . 7  |-  ( b  =  ( F `  a )  <->  ( F `  a )  =  b )
1817anbi2i 675 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( ( a  e.  A  /\  b  e.  B )  /\  ( F `  a )  =  b ) )
1915, 16, 183bitr2i 264 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
2013, 19bitr3i 242 . . . 4  |-  ( ( ( a  e.  A  /\  b  =  ( F `  a )
)  /\  ( a  e.  A  /\  b  e.  B ) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
2120opabbii 4084 . . 3  |-  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  =  ( F `  a )
)  /\  ( a  e.  A  /\  b  e.  B ) ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
2211, 12, 213eqtri 2308 . 2  |-  ( ( a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) )  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
238, 22syl6eq 2332 1  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685    i^i cin 3152    C_ wss 3153   {copab 4077    e. cmpt 4078    X. cxp 4686    Fn wfn 5216   -->wf 5217   ` cfv 5221
This theorem is referenced by:  fgraphxp  26941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229
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