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Theorem fgraphopab 31353
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 5749 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
2 df-ss 3485 . . . 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 5737 . . . . 5  |-  ( F : A --> B  ->  F  Fn  A )
5 dffn5 5918 . . . . 5  |-  ( F  Fn  A  <->  F  =  ( a  e.  A  |->  ( F `  a
) ) )
64, 5sylib 196 . . . 4  |-  ( F : A --> B  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
76ineq1d 3695 . . 3  |-  ( F : A --> B  -> 
( F  i^i  ( A  X.  B ) )  =  ( ( a  e.  A  |->  ( F `
 a ) )  i^i  ( A  X.  B ) ) )
83, 7eqtr3d 2500 . 2  |-  ( F : A --> B  ->  F  =  ( (
a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) ) )
9 df-mpt 4517 . . . 4  |-  ( a  e.  A  |->  ( F `
 a ) )  =  { <. a ,  b >.  |  ( a  e.  A  /\  b  =  ( F `  a ) ) }
10 df-xp 5014 . . . 4  |-  ( A  X.  B )  =  { <. a ,  b
>.  |  ( a  e.  A  /\  b  e.  B ) }
119, 10ineq12i 3694 . . 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 5143 . . 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 828 . . . . 5  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( (
a  e.  A  /\  b  =  ( F `  a ) )  /\  ( a  e.  A  /\  b  e.  B
) ) )
14 ancom 450 . . . . . . 7  |-  ( ( b  =  ( F `
 a )  /\  b  e.  B )  <->  ( b  e.  B  /\  b  =  ( F `  a ) ) )
1514anbi2i 694 . . . . . 6  |-  ( ( a  e.  A  /\  ( b  =  ( F `  a )  /\  b  e.  B
) )  <->  ( a  e.  A  /\  (
b  e.  B  /\  b  =  ( F `  a ) ) ) )
16 anass 649 . . . . . 6  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  b  =  ( F `  a ) )  <->  ( a  e.  A  /\  ( b  e.  B  /\  b  =  ( F `  a ) ) ) )
17 eqcom 2466 . . . . . . 7  |-  ( b  =  ( F `  a )  <->  ( F `  a )  =  b )
1817anbi2i 694 . . . . . 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 4521 . . 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 2490 . 2  |-  ( ( a  e.  A  |->  ( F `  a ) )  i^i  ( A  X.  B ) )  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
238, 22syl6eq 2514 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 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   {copab 4514    |-> cmpt 4515    X. cxp 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  fgraphxp  31354
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