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Theorem ofval 6103
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 2297 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2297 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6101 . . . 4  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5543 . . 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 5541 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
12 fveq2 5541 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1311, 12oveq12d 5892 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
14 eqid 2296 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
15 ovex 5899 . . . 4  |-  ( ( F `  X ) R ( G `  X ) )  e. 
_V
1613, 14, 15fvmpt 5618 . . 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 3402 . . . . . 6  |-  ( A  i^i  B )  C_  A
195, 18eqsstr3i 3222 . . . . 5  |-  S  C_  A
2019sseli 3189 . . . 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 3403 . . . . . 6  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3222 . . . . 5  |-  S  C_  B
2524sseli 3189 . . . 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 5892 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
2910, 17, 283eqtrd 2332 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 1632    e. wcel 1696    i^i cin 3164    e. cmpt 4093    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  fnfvof  6106  offveq  6114  ofc1  6116  ofc2  6117  ofsubeq0  9759  ofnegsub  9760  ofsubge0  9761  seqof  11119  o1of2  12102  gsumzaddlem  15219  psrbagcon  16133  psrbagconf1o  16136  psrdi  16167  psrdir  16168  mplsubglem  16195  mbfaddlem  19031  i1faddlem  19064  i1fmullem  19065  itg1lea  19083  mbfi1flimlem  19093  itg2split  19120  itg2monolem1  19121  itg2addlem  19129  dvaddbr  19303  dvmulbr  19304  plyaddlem1  19611  coeeulem  19622  coeaddlem  19646  dgradd2  19665  dgrcolem2  19671  ofmulrt  19678  plydivlem3  19691  plydivlem4  19692  plydiveu  19694  plyrem  19701  vieta1lem2  19707  elqaalem3  19717  qaa  19719  basellem7  20340  basellem9  20342  itg2addnc  25005  dgrsub2  27442  mpaaeu  27458  caofcan  27643  ofmul12  27645  ofdivrec  27646  ofdivcan4  27647  ofdivdiv2  27648  lfladdcl  29883  ldualvaddval  29943
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-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-pr 4230
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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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