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Theorem fgraphxp 26942
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 26941 . 2  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
2 vex 2791 . . . . . . 7  |-  a  e. 
_V
3 vex 2791 . . . . . . 7  |-  b  e. 
_V
42, 3op1std 6130 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
54fveq2d 5529 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( F `  ( 1st `  x ) )  =  ( F `
 a ) )
62, 3op2ndd 6131 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
75, 6eqeq12d 2297 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( F `
 ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( F `  a
)  =  b ) )
87rabxp 4725 . . 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 4083 . . 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 2303 . 2  |-  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
121, 11syl6eqr 2333 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 1623    e. wcel 1684   {crab 2547   <.cop 3643   {copab 4076    X. cxp 4687   -->wf 5251   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  hausgraph  26943
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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  df-1st 6122  df-2nd 6123
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