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Theorem fgraphxp 31333
Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
fgraphxp  |-  ( F : A --> B  ->  F  =  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) } )
Distinct variable groups:    x, F    x, A    x, B

Proof of Theorem fgraphxp
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fgraphopab 31332 . 2  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
2 vex 3112 . . . . . . 7  |-  a  e. 
_V
3 vex 3112 . . . . . . 7  |-  b  e. 
_V
42, 3op1std 6809 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
54fveq2d 5876 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( F `  ( 1st `  x ) )  =  ( F `
 a ) )
62, 3op2ndd 6810 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
75, 6eqeq12d 2479 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( F `
 ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( F `  a
)  =  b ) )
87rabxp 5045 . . 3  |-  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) }  =  { <. a ,  b >.  |  ( a  e.  A  /\  b  e.  B  /\  ( F `  a )  =  b ) }
9 df-3an 975 . . . 4  |-  ( ( a  e.  A  /\  b  e.  B  /\  ( F `  a )  =  b )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
109opabbii 4521 . . 3  |-  { <. a ,  b >.  |  ( a  e.  A  /\  b  e.  B  /\  ( F `  a )  =  b ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
118, 10eqtri 2486 . 2  |-  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
121, 11syl6eqr 2516 1  |-  ( F : A --> B  ->  F  =  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {crab 2811   <.cop 4038   {copab 4514    X. cxp 5006   -->wf 5590   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
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-8 1821  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-pow 4634  ax-pr 4695  ax-un 6591
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  df-1st 6799  df-2nd 6800
This theorem is referenced by:  hausgraph  31334
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