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Theorem fgraphopab 29503
Description: Express a function as a subset of the Cartesian 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 5567 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
2 df-ss 3339 . . . 4  |-  ( F 
C_  ( A  X.  B )  <->  ( F  i^i  ( A  X.  B
) )  =  F )
31, 2sylib 196 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  F )
4 ffn 5556 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
5 dffn5 5734 . . . . 5  |-  ( F  Fn  A  <->  F  =  ( a  e.  A  |->  ( F `  a
) ) )
64, 5sylib 196 . . . 4  |-  ( F : A --> B  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
76ineq1d 3548 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  ( ( a  e.  A  |->  ( F `
 a ) )  i^i  ( A  X.  B ) ) )
83, 7eqtr3d 2475 . 2  |-  ( F : A --> B  ->  F  =  ( (
a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) ) )
9 df-mpt 4349 . . . 4  |-  ( a  e.  A  |->  ( F `
 a ) )  =  { <. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a ) ) }
10 df-xp 4842 . . . 4  |-  ( A  X.  B )  =  { <. a ,  b
>.  |  ( a  e.  A  /\  b  e.  B ) }
119, 10ineq12i 3547 . . 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 4966 . . 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 819 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  =  ( F `  a ) )  /\  ( a  e.  A  /\  b  e.  B
) ) )
14 ancom 448 . . . . . . 7  |-  ( ( b  =  ( F `
 a )  /\  b  e.  B )  <->  ( b  e.  B  /\  b  =  ( F `  a ) ) )
1514anbi2i 689 . . . . . 6  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( a  e.  A  /\  (
b  e.  B  /\  b  =  ( F `  a ) ) ) )
16 anass 644 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( a  e.  A  /\  ( b  e.  B  /\  b  =  ( F `  a ) ) ) )
17 eqcom 2443 . . . . . . 7  |-  ( b  =  ( F `  a )  <->  ( F `  a )  =  b )
1817anbi2i 689 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( ( a  e.  A  /\  b  e.  B )  /\  ( F `  a )  =  b ) )
1915, 16, 183bitr2i 273 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
2013, 19bitr3i 251 . . . 4  |-  ( ( ( a  e.  A  /\  b  =  ( F `  a )
)  /\  ( a  e.  A  /\  b  e.  B ) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
2120opabbii 4353 . . 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 2465 . 2  |-  ( ( a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) )  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
238, 22syl6eq 2489 1  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325   {copab 4346    e. cmpt 4347    X. cxp 4834    Fn wfn 5410   -->wf 5411   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423
This theorem is referenced by:  fgraphxp  29504
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