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Theorem fgraphxp 27488
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 27487 . 2  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
2 vex 2951 . . . . . . 7  |-  a  e. 
_V
3 vex 2951 . . . . . . 7  |-  b  e. 
_V
42, 3op1std 6349 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
54fveq2d 5724 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( F `  ( 1st `  x ) )  =  ( F `
 a ) )
62, 3op2ndd 6350 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
75, 6eqeq12d 2449 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( F `
 ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( F `  a
)  =  b ) )
87rabxp 4906 . . 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 938 . . . 4  |-  ( ( a  e.  A  /\  b  e.  B  /\  ( F `  a )  =  b )  <->  ( (
a  e.  A  /\  b  e.  B )  /\  ( F `  a
)  =  b ) )
109opabbii 4264 . . 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 2455 . 2  |-  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
121, 11syl6eqr 2485 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701   <.cop 3809   {copab 4257    X. cxp 4868   -->wf 5442   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  hausgraph  27489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-1st 6341  df-2nd 6342
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