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Theorem offval 6085
Description: Value of an operation applied to two functions. (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
offval.6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
offval.7  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
Assertion
Ref Expression
offval  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Distinct variable groups:    x, A    x, F    x, G    ph, x    x, S    x, R
Allowed substitution hints:    B( x)    C( x)    D( x)    V( x)    W( x)

Proof of Theorem offval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4  |-  ( ph  ->  F  Fn  A )
2 offval.3 . . . 4  |-  ( ph  ->  A  e.  V )
3 fnex 5741 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 642 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 5741 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 642 . . 3  |-  ( ph  ->  G  e.  _V )
9 fndm 5343 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
101, 9syl 15 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
11 fndm 5343 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
125, 11syl 15 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
1310, 12ineq12d 3371 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
14 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
1513, 14syl6eq 2331 . . . . 5  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
16 mpteq1 4100 . . . . 5  |-  ( ( dom  F  i^i  dom  G )  =  S  -> 
( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
1715, 16syl 15 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
18 inex1g 4157 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
1914, 18syl5eqelr 2368 . . . . 5  |-  ( A  e.  V  ->  S  e.  _V )
20 mptexg 5745 . . . . 5  |-  ( S  e.  _V  ->  (
x  e.  S  |->  ( ( F `  x
) R ( G `
 x ) ) )  e.  _V )
212, 19, 203syl 18 . . . 4  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
2217, 21eqeltrd 2357 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
23 dmeq 4879 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
24 dmeq 4879 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
2523, 24ineqan12d 3372 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
26 fveq1 5524 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
27 fveq1 5524 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
2826, 27oveqan12d 5877 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
2925, 28mpteq12dv 4098 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x ) R ( g `  x ) ) )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x ) R ( G `  x ) ) ) )
30 df-of 6078 . . . 4  |-  o F R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
3129, 30ovmpt2ga 5977 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V  /\  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )  ->  ( F  o F R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
324, 8, 22, 31syl3anc 1182 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
3314eleq2i 2347 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  S
)
34 elin 3358 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3533, 34bitr3i 242 . . . 4  |-  ( x  e.  S  <->  ( x  e.  A  /\  x  e.  B ) )
36 offval.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3736adantrr 697 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( F `  x
)  =  C )
38 offval.7 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3938adantrl 696 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( G `  x
)  =  D )
4037, 39oveq12d 5876 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( ( F `  x ) R ( G `  x ) )  =  ( C R D ) )
4135, 40sylan2b 461 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( C R D ) )
4241mpteq2dva 4106 . 2  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( C R D ) ) )
4332, 17, 423eqtrd 2319 1  |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    e. cmpt 4077   dom cdm 4689    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  ofval  6087  offn  6089  off  6093  ofres  6094  offval2  6095  ofco  6097  offveqb  6099  suppssof1  6119  o1rlimmul  12092  gsumbagdiaglem  16121  psrplusgpropd  16313  mbfadd  19016  mbfsub  19017  mbfmullem2  19079  mbfmul  19081  bddmulibl  19193  dvcmulf  19294  evlslem1  19399  ofrn2  23207  off2  23208  offval2f  23233
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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