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Theorem ofval 6087
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  o F 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 2284 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2284 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6085 . . . 4  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5527 . . 3  |-  ( ph  ->  ( ( F  o F R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
109adantr 451 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  o F R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
11 fveq2 5525 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
12 fveq2 5525 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1311, 12oveq12d 5876 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
14 eqid 2283 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
15 ovex 5883 . . . 4  |-  ( ( F `  X ) R ( G `  X ) )  e. 
_V
1613, 14, 15fvmpt 5602 . . 3  |-  ( X  e.  S  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
1716adantl 452 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
18 inss1 3389 . . . . . 6  |-  ( A  i^i  B )  C_  A
195, 18eqsstr3i 3209 . . . . 5  |-  S  C_  A
2019sseli 3176 . . . 4  |-  ( X  e.  S  ->  X  e.  A )
21 ofval.6 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2220, 21sylan2 460 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  =  C )
23 inss2 3390 . . . . . 6  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3209 . . . . 5  |-  S  C_  B
2524sseli 3176 . . . 4  |-  ( X  e.  S  ->  X  e.  B )
26 ofval.7 . . . 4  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2725, 26sylan2 460 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  =  D )
2822, 27oveq12d 5876 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
2910, 17, 283eqtrd 2319 1  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  o F R G ) `  X )  =  ( C R D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    e. cmpt 4077    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  fnfvof  6090  offveq  6098  ofc1  6100  ofc2  6101  ofsubeq0  9743  ofnegsub  9744  ofsubge0  9745  seqof  11103  o1of2  12086  gsumzaddlem  15203  psrbagcon  16117  psrbagconf1o  16120  psrdi  16151  psrdir  16152  mplsubglem  16179  mbfaddlem  19015  i1faddlem  19048  i1fmullem  19049  itg1lea  19067  mbfi1flimlem  19077  itg2split  19104  itg2monolem1  19105  itg2addlem  19113  dvaddbr  19287  dvmulbr  19288  plyaddlem1  19595  coeeulem  19606  coeaddlem  19630  dgradd2  19649  dgrcolem2  19655  ofmulrt  19662  plydivlem3  19675  plydivlem4  19676  plydiveu  19678  plyrem  19685  vieta1lem2  19691  elqaalem3  19701  qaa  19703  basellem7  20324  basellem9  20326  dgrsub2  27339  mpaaeu  27355  caofcan  27540  ofmul12  27542  ofdivrec  27543  ofdivcan4  27544  ofdivdiv2  27545  lfladdcl  29261  ldualvaddval  29321
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-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
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-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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