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Theorem fgraphxp 27633
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 27632 . 2  |-  ( F : A --> B  ->  F  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) } )
2 vex 2804 . . . . . . 7  |-  a  e. 
_V
3 vex 2804 . . . . . . 7  |-  b  e. 
_V
42, 3op1std 6146 . . . . . 6  |-  ( x  =  <. a ,  b
>.  ->  ( 1st `  x
)  =  a )
54fveq2d 5545 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( F `  ( 1st `  x ) )  =  ( F `
 a ) )
62, 3op2ndd 6147 . . . . 5  |-  ( x  =  <. a ,  b
>.  ->  ( 2nd `  x
)  =  b )
75, 6eqeq12d 2310 . . . 4  |-  ( x  =  <. a ,  b
>.  ->  ( ( F `
 ( 1st `  x
) )  =  ( 2nd `  x )  <-> 
( F `  a
)  =  b ) )
87rabxp 4741 . . 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 4099 . . 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 2316 . 2  |-  { x  e.  ( A  X.  B
)  |  ( F `
 ( 1st `  x
) )  =  ( 2nd `  x ) }  =  { <. a ,  b >.  |  ( ( a  e.  A  /\  b  e.  B
)  /\  ( F `  a )  =  b ) }
121, 11syl6eqr 2346 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 1632    e. wcel 1696   {crab 2560   <.cop 3656   {copab 4092    X. cxp 4703   -->wf 5267   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  hausgraph  27634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-1st 6138  df-2nd 6139
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