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Theorem offval 5877
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  oF 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 5533 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 404 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 5533 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 404 . . 3  |-  ( ph  ->  G  e.  _V )
9 fndm 5126 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
101, 9syl 14 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
11 fndm 5126 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
125, 11syl 14 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
1310, 12ineq12d 3203 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
14 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
1513, 14syl6eq 2137 . . . . 5  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
1615mpteq1d 3929 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
17 inex1g 3981 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
1814, 17syl5eqelr 2176 . . . . 5  |-  ( A  e.  V  ->  S  e.  _V )
19 mptexg 5536 . . . . 5  |-  ( S  e.  _V  ->  (
x  e.  S  |->  ( ( F `  x
) R ( G `
 x ) ) )  e.  _V )
202, 18, 193syl 17 . . . 4  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
2116, 20eqeltrd 2165 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
22 dmeq 4649 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
23 dmeq 4649 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
2422, 23ineqan12d 3204 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
25 fveq1 5317 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
26 fveq1 5317 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
2725, 26oveqan12d 5685 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
2824, 27mpteq12dv 3926 . . . 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 ) ) ) )
29 df-of 5870 . . . 4  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
3028, 29ovmpt2ga 5788 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V  /\  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x ) R ( G `  x ) ) ) )
314, 8, 21, 30syl3anc 1175 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
3214eleq2i 2155 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  S
)
33 elin 3184 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3432, 33bitr3i 185 . . . 4  |-  ( x  e.  S  <->  ( x  e.  A  /\  x  e.  B ) )
35 offval.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3635adantrr 464 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( F `  x
)  =  C )
37 offval.7 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3837adantrl 463 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( G `  x
)  =  D )
3936, 38oveq12d 5684 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( ( F `  x ) R ( G `  x ) )  =  ( C R D ) )
4034, 39sylan2b 282 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( C R D ) )
4140mpteq2dva 3934 . 2  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( C R D ) ) )
4231, 16, 413eqtrd 2125 1  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   _Vcvv 2620    i^i cin 2999    |-> cmpt 3905   dom cdm 4452    Fn wfn 5023   ` cfv 5028  (class class class)co 5666    oFcof 5868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-of 5870
This theorem is referenced by:  fnofval  5879  off  5882  ofres  5883  offval2  5884  suppssof1  5886  ofco  5887  offveqb  5888
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