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Theorem offval 6101
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 5757 . . . 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 5757 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 642 . . 3  |-  ( ph  ->  G  e.  _V )
9 fndm 5359 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
101, 9syl 15 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
11 fndm 5359 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
125, 11syl 15 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
1310, 12ineq12d 3384 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
14 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
1513, 14syl6eq 2344 . . . . 5  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
16 mpteq1 4116 . . . . 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 4173 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
1914, 18syl5eqelr 2381 . . . . 5  |-  ( A  e.  V  ->  S  e.  _V )
20 mptexg 5761 . . . . 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 2370 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
23 dmeq 4895 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
24 dmeq 4895 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
2523, 24ineqan12d 3385 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
26 fveq1 5540 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
27 fveq1 5540 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
2826, 27oveqan12d 5893 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
2925, 28mpteq12dv 4114 . . . 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 6094 . . . 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 5993 . . 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 2360 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  S
)
34 elin 3371 . . . . 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 5892 . . . 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 4122 . 2  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( C R D ) ) )
4332, 17, 423eqtrd 2332 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    e. cmpt 4093   dom cdm 4705    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  ofval  6103  offn  6105  off  6109  ofres  6110  offval2  6111  ofco  6113  offveqb  6115  suppssof1  6135  o1rlimmul  12108  gsumbagdiaglem  16137  psrplusgpropd  16329  mbfadd  19032  mbfsub  19033  mbfmullem2  19095  mbfmul  19097  bddmulibl  19209  dvcmulf  19310  evlslem1  19415  ofrn2  23222  off2  23223  offval2f  23248  itg2addnclem  25003  itg2addnc  25005
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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