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Theorem ofval 6253
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 2388 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2388 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6251 . . . 4  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5670 . . 3  |-  ( ph  ->  ( ( F  o F R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
109adantr 452 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  o F R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
11 fveq2 5668 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
12 fveq2 5668 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1311, 12oveq12d 6038 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
14 eqid 2387 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
15 ovex 6045 . . . 4  |-  ( ( F `  X ) R ( G `  X ) )  e. 
_V
1613, 14, 15fvmpt 5745 . . 3  |-  ( X  e.  S  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
1716adantl 453 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
18 inss1 3504 . . . . . 6  |-  ( A  i^i  B )  C_  A
195, 18eqsstr3i 3322 . . . . 5  |-  S  C_  A
2019sseli 3287 . . . 4  |-  ( X  e.  S  ->  X  e.  A )
21 ofval.6 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2220, 21sylan2 461 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  =  C )
23 inss2 3505 . . . . . 6  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3322 . . . . 5  |-  S  C_  B
2524sseli 3287 . . . 4  |-  ( X  e.  S  ->  X  e.  B )
26 ofval.7 . . . 4  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2725, 26sylan2 461 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  =  D )
2822, 27oveq12d 6038 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
2910, 17, 283eqtrd 2423 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 359    = wceq 1649    e. wcel 1717    i^i cin 3262    e. cmpt 4207    Fn wfn 5389   ` cfv 5394  (class class class)co 6020    o Fcof 6242
This theorem is referenced by:  fnfvof  6256  offveq  6264  ofc1  6266  ofc2  6267  ofsubeq0  9929  ofnegsub  9930  ofsubge0  9931  seqof  11307  o1of2  12333  gsumzaddlem  15453  psrbagcon  16363  psrbagconf1o  16366  psrdi  16397  psrdir  16398  mplsubglem  16425  mbfaddlem  19419  i1faddlem  19452  i1fmullem  19453  itg1lea  19471  mbfi1flimlem  19481  itg2split  19508  itg2monolem1  19509  itg2addlem  19517  dvaddbr  19691  dvmulbr  19692  plyaddlem1  19999  coeeulem  20010  coeaddlem  20034  dgradd2  20053  dgrcolem2  20059  ofmulrt  20066  plydivlem3  20079  plydivlem4  20080  plydiveu  20082  plyrem  20089  vieta1lem2  20095  elqaalem3  20105  qaa  20107  basellem7  20736  basellem9  20738  itg2addnc  25959  dgrsub2  27008  mpaaeu  27024  caofcan  27209  ofmul12  27211  ofdivrec  27212  ofdivcan4  27213  ofdivdiv2  27214  lfladdcl  29186  ldualvaddval  29246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244
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