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Theorem ofval 6554
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 2430 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2430 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7offval 6552 . . . 4  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
98fveq1d 5883 . . 3  |-  ( ph  ->  ( ( F  oF R G ) `
 X )  =  ( ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) ) `  X ) )
109adantr 466 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
) )
11 fveq2 5881 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
12 fveq2 5881 . . . . 5  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1311, 12oveq12d 6323 . . . 4  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x ) )  =  ( ( F `
 X ) R ( G `  X
) ) )
14 eqid 2429 . . . 4  |-  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( ( F `
 x ) R ( G `  x
) ) )
15 ovex 6333 . . . 4  |-  ( ( F `  X ) R ( G `  X ) )  e. 
_V
1613, 14, 15fvmpt 5964 . . 3  |-  ( X  e.  S  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
1716adantl 467 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) `  X
)  =  ( ( F `  X ) R ( G `  X ) ) )
18 inss1 3688 . . . . . 6  |-  ( A  i^i  B )  C_  A
195, 18eqsstr3i 3501 . . . . 5  |-  S  C_  A
2019sseli 3466 . . . 4  |-  ( X  e.  S  ->  X  e.  A )
21 ofval.6 . . . 4  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2220, 21sylan2 476 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( F `  X )  =  C )
23 inss2 3689 . . . . . 6  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3501 . . . . 5  |-  S  C_  B
2524sseli 3466 . . . 4  |-  ( X  e.  S  ->  X  e.  B )
26 ofval.7 . . . 4  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2725, 26sylan2 476 . . 3  |-  ( (
ph  /\  X  e.  S )  ->  ( G `  X )  =  D )
2822, 27oveq12d 6323 . 2  |-  ( (
ph  /\  X  e.  S )  ->  (
( F `  X
) R ( G `
 X ) )  =  ( C R D ) )
2910, 17, 283eqtrd 2474 1  |-  ( (
ph  /\  X  e.  S )  ->  (
( F  oF R G ) `  X )  =  ( C R D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    i^i cin 3441    |-> cmpt 4484    Fn wfn 5596   ` cfv 5601  (class class class)co 6305    oFcof 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545
This theorem is referenced by:  fnfvof  6559  offveq  6566  ofc1  6568  ofc2  6569  suppofss1d  6963  suppofss2d  6964  ofsubeq0  10606  ofnegsub  10607  ofsubge0  10608  seqof  12267  o1of2  13654  gsumzaddlem  17489  psrbagcon  18530  psrbagconf1o  18533  psrdi  18565  psrdir  18566  mplsubglem  18593  matplusgcell  19389  matsubgcell  19390  rrxcph  22244  mbfaddlem  22493  i1faddlem  22528  i1fmullem  22529  itg1lea  22547  mbfi1flimlem  22557  itg2split  22584  itg2monolem1  22585  itg2addlem  22593  dvaddbr  22769  dvmulbr  22770  plyaddlem1  23035  coeeulem  23046  coeaddlem  23071  dgradd2  23090  dgrcolem2  23096  ofmulrt  23103  plydivlem3  23116  plydivlem4  23117  plydiveu  23119  plyrem  23126  vieta1lem2  23132  elqaalem3  23142  qaa  23144  basellem7  23876  basellem9  23878  poimirlem1  31645  poimirlem2  31646  poimirlem6  31650  poimirlem7  31651  poimirlem10  31654  poimirlem11  31655  poimirlem12  31656  poimirlem17  31661  poimirlem20  31664  poimirlem23  31667  poimirlem29  31673  poimirlem31  31675  poimirlem32  31676  broucube  31678  itg2addnclem3  31699  itg2addnc  31700  ftc1anclem5  31725  lfladdcl  32346  ldualvaddval  32406  dgrsub2  35700  mpaaeu  35715  caofcan  36309  ofmul12  36311  ofdivrec  36312  ofdivcan4  36313  ofdivdiv2  36314  binomcxplemrat  36336  binomcxplemnotnn0  36342  mndpsuppss  38925
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