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Theorem fgraphopab 26695
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 5257 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
2 df-ss 3089 . . . 4  |-  ( F 
C_  ( A  X.  B )  <->  ( F  i^i  ( A  X.  B
) )  =  F )
31, 2sylib 190 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  F )
4 ffn 5246 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
5 dffn5 5420 . . . . 5  |-  ( F  Fn  A  <->  F  =  ( a  e.  A  |->  ( F `  a
) ) )
64, 5sylib 190 . . . 4  |-  ( F : A --> B  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
76ineq1d 3277 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  ( ( a  e.  A  |->  ( F `
 a ) )  i^i  ( A  X.  B ) ) )
83, 7eqtr3d 2287 . 2  |-  ( F : A --> B  ->  F  =  ( (
a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) ) )
9 df-mpt 3976 . . . 4  |-  ( a  e.  A  |->  ( F `
 a ) )  =  { <. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a ) ) }
10 df-xp 4594 . . . 4  |-  ( A  X.  B )  =  { <. a ,  b
>.  |  ( a  e.  A  /\  b  e.  B ) }
119, 10ineq12i 3276 . . 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 4723 . . 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 804 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  =  ( F `  a ) )  /\  ( a  e.  A  /\  b  e.  B
) ) )
14 ancom 439 . . . . . . 7  |-  ( ( b  =  ( F `
 a )  /\  b  e.  B )  <->  ( b  e.  B  /\  b  =  ( F `  a ) ) )
1514anbi2i 678 . . . . . 6  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( a  e.  A  /\  (
b  e.  B  /\  b  =  ( F `  a ) ) ) )
16 anass 633 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( a  e.  A  /\  ( b  e.  B  /\  b  =  ( F `  a ) ) ) )
17 eqcom 2255 . . . . . . 7  |-  ( b  =  ( F `  a )  <->  ( F `  a )  =  b )
1817anbi2i 678 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( ( a  e.  A  /\  b  e.  B )  /\  ( F `  a )  =  b ) )
1915, 16, 183bitr2i 266 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
2013, 19bitr3i 244 . . . 4  |-  ( ( ( a  e.  A  /\  b  =  ( F `  a )
)  /\  ( a  e.  A  /\  b  e.  B ) )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
2120opabbii 3980 . . 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 2277 . 2  |-  ( ( a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) )  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
238, 22syl6eq 2301 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 6    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3077    C_ wss 3078   {copab 3973    e. cmpt 3974    X. cxp 4578    Fn wfn 4587   -->wf 4588   ` cfv 4592
This theorem is referenced by:  fgraphxp  26696
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608
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