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Theorem cur1vald 24611
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 4800 . . . . . . . 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 5741 . . . . . . 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 5341 . . . 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 5343 . . . . 5  |-  ( F  Fn  ( A  X.  B )  ->  dom  F  =  ( A  X.  B ) )
12 relxp 4794 . . . . . 6  |-  Rel  ( A  X.  B )
13 releq 4771 . . . . . . 7  |-  ( ( A  X.  B )  =  dom  F  -> 
( Rel  ( A  X.  B )  <->  Rel  dom  F
) )
1413eqcoms 2286 . . . . . 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 24610 . . 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 4881 . . . . . 6  |-  ( F  Fn  ( A  X.  B )  ->  dom  dom 
F  =  dom  ( A  X.  B ) )
21 dmxp 4897 . . . . . . . . 9  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
2221eqeq1d 2291 . . . . . . . 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 2286 . . . . . 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 4100 . . 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 2318 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   2ndc2nd 6121   cur1ccur1 24606
This theorem is referenced by:  domrancur1b  24612  domrancur1c  24614
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-pow 4188  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-pw 3627  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 24608
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