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Theorem ofval 6533
Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1  |-  ( ph  ->  F  Fn  A )
offval.2  |-  ( ph  ->  G  Fn  B )
offval.3  |-  ( ph  ->  A  e.  V )
offval.4  |-  ( ph  ->  B  e.  W )
offval.5  |-  ( A  i^i  B )  =  S
ofval.6  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
ofval.7  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
Assertion
Ref Expression
ofval  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( C R D ) )

Proof of Theorem ofval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . 5  |-  ( ph  ->  F  Fn  A )
2 offval.2 . . . . 5  |-  ( ph  ->  G  Fn  B )
3 offval.3 . . . . 5  |-  ( ph  ->  A  e.  V )
4 offval.4 . . . . 5  |-  ( ph  ->  B  e.  W )
5 offval.5 . . . . 5  |-  ( A  i^i  B )  =  S
6 eqidd 2468 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2468 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6531 . . . 4  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5868 . . 3  |-  ( ph  ->  ( ( F  oF R G ) `
 X )  =  ( ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) ) `  X ) )
109adantr 465 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
11 fveq2 5866 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
12 fveq2 5866 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1311, 12oveq12d 6302 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
14 eqid 2467 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
15 ovex 6309 . . . 4  |-  ( ( F `  X ) R ( G `  X ) )  e. 
_V
1613, 14, 15fvmpt 5950 . . 3  |-  ( X  e.  S  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
1716adantl 466 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
18 inss1 3718 . . . . . 6  |-  ( A  i^i  B )  C_  A
195, 18eqsstr3i 3535 . . . . 5  |-  S  C_  A
2019sseli 3500 . . . 4  |-  ( X  e.  S  ->  X  e.  A )
21 ofval.6 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2220, 21sylan2 474 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  =  C )
23 inss2 3719 . . . . . 6  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3535 . . . . 5  |-  S  C_  B
2524sseli 3500 . . . 4  |-  ( X  e.  S  ->  X  e.  B )
26 ofval.7 . . . 4  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2725, 26sylan2 474 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  =  D )
2822, 27oveq12d 6302 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
2910, 17, 283eqtrd 2512 1  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( C R D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3475    |-> cmpt 4505    Fn wfn 5583   ` cfv 5588  (class class class)co 6284    oFcof 6522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524
This theorem is referenced by:  fnfvof  6537  offveq  6545  ofc1  6547  ofc2  6548  suppofss1d  6937  suppofss2d  6938  ofsubeq0  10533  ofnegsub  10534  ofsubge0  10535  seqof  12132  o1of2  13398  gsumzaddlem  16737  gsumzaddlemOLD  16739  psrbagcon  17821  psrbagconf1o  17825  psrdi  17860  psrdir  17861  mplsubglem  17892  mplsubglemOLD  17894  matplusgcell  18730  matsubgcell  18731  rrxcph  21587  mbfaddlem  21830  i1faddlem  21863  i1fmullem  21864  itg1lea  21882  mbfi1flimlem  21892  itg2split  21919  itg2monolem1  21920  itg2addlem  21928  dvaddbr  22104  dvmulbr  22105  plyaddlem1  22373  coeeulem  22384  coeaddlem  22408  dgradd2  22427  dgrcolem2  22433  ofmulrt  22440  plydivlem3  22453  plydivlem4  22454  plydiveu  22456  plyrem  22463  vieta1lem2  22469  elqaalem3  22479  qaa  22481  basellem7  23116  basellem9  23118  itg2addnclem3  29673  itg2addnc  29674  ftc1anclem5  29699  dgrsub2  30716  mpaaeu  30732  caofcan  30856  ofmul12  30858  ofdivrec  30859  ofdivcan4  30860  ofdivdiv2  30861  mndpsuppss  32063  lfladdcl  33886  ldualvaddval  33946
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