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Theorem fgraphopab 27529
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 5400 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
2 df-ss 3166 . . . 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 5389 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
5 dffn5 5568 . . . . 5  |-  ( F  Fn  A  <->  F  =  ( a  e.  A  |->  ( F `  a
) ) )
64, 5sylib 188 . . . 4  |-  ( F : A --> B  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
76ineq1d 3369 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  ( ( a  e.  A  |->  ( F `
 a ) )  i^i  ( A  X.  B ) ) )
83, 7eqtr3d 2317 . 2  |-  ( F : A --> B  ->  F  =  ( (
a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) ) )
9 df-mpt 4079 . . . 4  |-  ( a  e.  A  |->  ( F `
 a ) )  =  { <. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a ) ) }
10 df-xp 4695 . . . 4  |-  ( A  X.  B )  =  { <. a ,  b
>.  |  ( a  e.  A  /\  b  e.  B ) }
119, 10ineq12i 3368 . . 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 4816 . . 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 2285 . . . . . . 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 4083 . . 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 2307 . 2  |-  ( ( a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) )  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
238, 22syl6eq 2331 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 1684    i^i cin 3151    C_ wss 3152   {copab 4076    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  fgraphxp  27530
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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