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Theorem fgraphxp 26853
Description: Express a function as a subset of the cross 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 26852 . 2  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
2 vex 2867 . . . . . . 7  |-  a  e. 
_V
3 vex 2867 . . . . . . 7  |-  b  e. 
_V
42, 3op1std 6214 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
54fveq2d 5609 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( F `  ( 1st `  x ) )  =  ( F `
 a ) )
62, 3op2ndd 6215 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
75, 6eqeq12d 2372 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( F `
 ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( F `  a
)  =  b ) )
87rabxp 4804 . . 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 936 . . . 4  |-  ( ( a  e.  A  /\  b  e.  B  /\  ( F `  a )  =  b )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
109opabbii 4162 . . 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 2378 . 2  |-  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
121, 11syl6eqr 2408 1  |-  ( F : A --> B  ->  F  =  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   {crab 2623   <.cop 3719   {copab 4155    X. cxp 4766   -->wf 5330   ` cfv 5334   1stc1st 6204   2ndc2nd 6205
This theorem is referenced by:  hausgraph  26854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-1st 6206  df-2nd 6207
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