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Theorem cur1val 25198
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 2796 . . 3  |-  ( F  e.  A  ->  F  e.  _V )
213ad2ant1 976 . 2  |-  ( ( F  e.  A  /\  Fun  F  /\  Rel  dom  F )  ->  F  e.  _V )
3 dmexg 4939 . . . . 5  |-  ( F  e.  A  ->  dom  F  e.  _V )
4 dmexg 4939 . . . . 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 5745 . . 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 25196 . . 3  |-  cur1  =  { <. f ,  g
>.  |  ( ( Fun  f  /\  Rel  dom  f )  /\  g  =  ( x  e. 
dom  dom  f  |->  ( f  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) ) }
11 funeq 5274 . . . 4  |-  ( f  =  F  ->  ( Fun  f  <->  Fun  F ) )
12 dmeq 4879 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
1312releqd 4773 . . . 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 4881 . . . 4  |-  ( f  =  F  ->  dom  dom  f  =  dom  dom  F )
16 coeq1 4841 . . . 4  |-  ( f  =  F  ->  (
f  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) )  =  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )
1715, 16mpteq12dv 4098 . . 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 5621 . 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249   ` cfv 5255   2ndc2nd 6121   cur1ccur1 25194
This theorem is referenced by:  cur1vald  25199  valcurfn1  25204
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-rep 4131  ax-sep 4141  ax-nul 4149  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-cur1 25196
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