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Theorem cur1vald 25302
Description: The value of a curried operation. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
cur1vald  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D    x, F

Proof of Theorem cur1vald
StepHypRef Expression
1 simp1 955 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  A  e.  C  /\  B  e.  D )  ->  F  Fn  ( A  X.  B ) )
2 xpexg 4816 . . . . . . . 8  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
323adant1 973 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
4 fnex 5757 . . . . . . 7  |-  ( ( F  Fn  ( A  X.  B )  /\  ( A  X.  B
)  e.  _V )  ->  F  e.  _V )
51, 3, 4syl2anc 642 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
653expib 1154 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  (
( A  e.  C  /\  B  e.  D
)  ->  F  e.  _V ) )
76adantr 451 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  ->  (
( A  e.  C  /\  B  e.  D
)  ->  F  e.  _V ) )
87imp 418 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  F  e.  _V )
9 fnfun 5357 . . . 4  |-  ( F  Fn  ( A  X.  B )  ->  Fun  F )
109ad2antrr 706 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  Fun  F )
11 fndm 5359 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 relxp 4810 . . . . . 6  |-  Rel  ( A  X.  B )
13 releq 4787 . . . . . . 7  |-  ( ( A  X.  B )  =  dom  F  -> 
( Rel  ( A  X.  B )  <->  Rel  dom  F
) )
1413eqcoms 2299 . . . . . 6  |-  ( dom 
F  =  ( A  X.  B )  -> 
( Rel  ( A  X.  B )  <->  Rel  dom  F
) )
1512, 14mpbii 202 . . . . 5  |-  ( dom 
F  =  ( A  X.  B )  ->  Rel  dom  F )
1611, 15syl 15 . . . 4  |-  ( F  Fn  ( A  X.  B )  ->  Rel  dom 
F )
1716ad2antrr 706 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  Rel  dom  F
)
18 cur1val 25301 . . 3  |-  ( ( F  e.  _V  /\  Fun  F  /\  Rel  dom  F )  ->  ( cur1 `  F )  =  ( x  e.  dom  dom  F 
|->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
198, 10, 17, 18syl3anc 1182 . 2  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( cur1 `  F )  =  ( x  e.  dom  dom  F 
|->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
2011dmeqd 4897 . . . . . 6  |-  ( F  Fn  ( A  X.  B )  ->  dom  dom 
F  =  dom  ( A  X.  B ) )
21 dmxp 4913 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
2221eqeq1d 2304 . . . . . . . 8  |-  ( B  =/=  (/)  ->  ( dom  ( A  X.  B
)  =  dom  dom  F  <-> 
A  =  dom  dom  F ) )
2322biimpcd 215 . . . . . . 7  |-  ( dom  ( A  X.  B
)  =  dom  dom  F  ->  ( B  =/=  (/)  ->  A  =  dom  dom 
F ) )
2423eqcoms 2299 . . . . . 6  |-  ( dom 
dom  F  =  dom  ( A  X.  B
)  ->  ( B  =/=  (/)  ->  A  =  dom  dom  F ) )
2520, 24syl 15 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  ( B  =/=  (/)  ->  A  =  dom  dom  F ) )
2625imp 418 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  ->  A  =  dom  dom  F )
2726adantr 451 . . 3  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  A  =  dom  dom  F )
28 mpteq1 4116 . . 3  |-  ( A  =  dom  dom  F  ->  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) )  =  ( x  e. 
dom  dom  F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
2927, 28syl 15 . 2  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) )  =  ( x  e.  dom  dom  F  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
3019, 29eqtr4d 2331 1  |-  ( ( ( F  Fn  ( A  X.  B )  /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( {
x }  X.  _V ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705    |` cres 4707    o. ccom 4709   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   ` cfv 5271   2ndc2nd 6137   cur1ccur1 25297
This theorem is referenced by:  domrancur1b  25303  domrancur1c  25305
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-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-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-pw 3640  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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