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Theorem cur1val 25301
Description: The value of a curried operation. (Contributed by FL, 24-Jan-2010.)
Assertion
Ref Expression
cur1val  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  ( cur1 `  F )  =  ( x  e.  dom  dom  F 
|->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
Distinct variable group:    x, F
Allowed substitution hint:    A( x)

Proof of Theorem cur1val
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2809 . . 3  |-  ( F  e.  A  ->  F  e.  _V )
213ad2ant1 976 . 2  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  F  e.  _V )
3 dmexg 4955 . . . . 5  |-  ( F  e.  A  ->  dom  F  e.  _V )
4 dmexg 4955 . . . . 5  |-  ( dom 
F  e.  _V  ->  dom 
dom  F  e.  _V )
53, 4syl 15 . . . 4  |-  ( F  e.  A  ->  dom  dom 
F  e.  _V )
653ad2ant1 976 . . 3  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  dom  dom  F  e.  _V )
7 mptexg 5761 . . 3  |-  ( dom 
dom  F  e.  _V  ->  ( x  e.  dom  dom 
F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )  e.  _V )
86, 7syl 15 . 2  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  ( x  e.  dom  dom  F  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )  e.  _V )
9 3simpc 954 . 2  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  ( Fun  F  /\  Rel  dom  F
) )
10 df-cur1 25299 . . 3  |-  cur1  =  { <. f ,  g
>.  |  ( ( Fun  f  /\  Rel  dom  f )  /\  g  =  ( x  e. 
dom  dom  f  |->  ( f  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) ) }
11 funeq 5290 . . . 4  |-  ( f  =  F  ->  ( Fun  f  <->  Fun  F ) )
12 dmeq 4895 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1312releqd 4789 . . . 4  |-  ( f  =  F  ->  ( Rel  dom  f  <->  Rel  dom  F
) )
1411, 13anbi12d 691 . . 3  |-  ( f  =  F  ->  (
( Fun  f  /\  Rel  dom  f )  <->  ( Fun  F  /\  Rel  dom  F
) ) )
1512dmeqd 4897 . . . 4  |-  ( f  =  F  ->  dom  dom  f  =  dom  dom  F )
16 coeq1 4857 . . . 4  |-  ( f  =  F  ->  (
f  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  =  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )
1715, 16mpteq12dv 4114 . . 3  |-  ( f  =  F  ->  (
x  e.  dom  dom  f  |->  ( f  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )  =  ( x  e. 
dom  dom  F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
1810, 14, 17fvopab6 5637 . 2  |-  ( ( F  e.  _V  /\  ( x  e.  dom  dom 
F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )  e.  _V  /\  ( Fun  F  /\  Rel  dom  F ) )  ->  ( cur1 `  F )  =  ( x  e.  dom  dom 
F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
192, 8, 9, 18syl3anc 1182 1  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  ( cur1 `  F )  =  ( x  e.  dom  dom  F 
|->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705    |` cres 4707    o. ccom 4709   Rel wrel 4710   Fun wfun 5265   ` cfv 5271   2ndc2nd 6137   cur1ccur1 25297
This theorem is referenced by:  cur1vald  25302  valcurfn1  25307
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-rep 4147  ax-sep 4157  ax-nul 4165  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-cur1 25299
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