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Theorem offval 6557
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 6148 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 673 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 6148 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 673 . . 3  |-  ( ph  ->  G  e.  _V )
9 fndm 5685 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
101, 9syl 17 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
11 fndm 5685 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
125, 11syl 17 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
1310, 12ineq12d 3626 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
14 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
1513, 14syl6eq 2521 . . . . 5  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
1615mpteq1d 4477 . . . 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 4539 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
1814, 17syl5eqelr 2554 . . . . 5  |-  ( A  e.  V  ->  S  e.  _V )
19 mptexg 6151 . . . . 5  |-  ( S  e.  _V  ->  (
x  e.  S  |->  ( ( F `  x
) R ( G `
 x ) ) )  e.  _V )
202, 18, 193syl 18 . . . 4  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
2116, 20eqeltrd 2549 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
22 dmeq 5040 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
23 dmeq 5040 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
2422, 23ineqan12d 3627 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
25 fveq1 5878 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
26 fveq1 5878 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
2725, 26oveqan12d 6327 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
2824, 27mpteq12dv 4474 . . . 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 6550 . . . 4  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
3028, 29ovmpt2ga 6445 . . 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 1292 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
3214eleq2i 2541 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  S
)
33 elin 3608 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3432, 33bitr3i 259 . . . 4  |-  ( x  e.  S  <->  ( x  e.  A  /\  x  e.  B ) )
35 offval.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3635adantrr 731 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( F `  x
)  =  C )
37 offval.7 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3837adantrl 730 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( G `  x
)  =  D )
3936, 38oveq12d 6326 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( ( F `  x ) R ( G `  x ) )  =  ( C R D ) )
4034, 39sylan2b 483 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( C R D ) )
4140mpteq2dva 4482 . 2  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( C R D ) ) )
4231, 16, 413eqtrd 2509 1  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    |-> cmpt 4454   dom cdm 4839    Fn wfn 5584   ` cfv 5589  (class class class)co 6308    oFcof 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550
This theorem is referenced by:  ofval  6559  offn  6561  offval2f  6562  off  6565  ofres  6566  offval2  6567  ofco  6570  offveqb  6572  suppssof1  6967  o1rlimmul  13759  gsumbagdiaglem  18676  evlslem1  18815  psrplusgpropd  18906  frlmipval  19414  frlmphllem  19415  frlmphl  19416  mat1dimscm  19577  rrxcph  22429  rrxds  22430  mbfadd  22696  mbfsub  22697  mbfmullem2  22761  mbfmul  22763  bddmulibl  22875  dvcmulf  22978  ofrn2  28317  off2  28318  ofresid  28319  ofcof  29002  plymul02  29507  signsplypnf  29511  signsply0  29512  poimirlem3  32007  poimirlem4  32008  poimirlem16  32020  poimirlem19  32023  poimirlem28  32032  broucube  32038  itg2addnc  32060  ftc1anclem8  32088  dflinc2  40711  fdivmpt  40859
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