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Theorem offval 5747
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 5411 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 397 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 5411 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 397 . . 3  |-  ( ph  ->  G  e.  _V )
9 fndm 5026 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
101, 9syl 14 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
11 fndm 5026 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
125, 11syl 14 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
1310, 12ineq12d 3167 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
14 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
1513, 14syl6eq 2104 . . . . 5  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
1615mpteq1d 3870 . . . 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 3921 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
1814, 17syl5eqelr 2141 . . . . 5  |-  ( A  e.  V  ->  S  e.  _V )
19 mptexg 5414 . . . . 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 2130 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
22 dmeq 4563 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
23 dmeq 4563 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
2422, 23ineqan12d 3168 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
25 fveq1 5205 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
26 fveq1 5205 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
2725, 26oveqan12d 5559 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
2824, 27mpteq12dv 3867 . . . 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 5740 . . . 4  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
3028, 29ovmpt2ga 5658 . . 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 1146 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
3214eleq2i 2120 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  S
)
33 elin 3154 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3432, 33bitr3i 179 . . . 4  |-  ( x  e.  S  <->  ( x  e.  A  /\  x  e.  B ) )
35 offval.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3635adantrr 456 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( F `  x
)  =  C )
37 offval.7 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3837adantrl 455 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( G `  x
)  =  D )
3936, 38oveq12d 5558 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( ( F `  x ) R ( G `  x ) )  =  ( C R D ) )
4034, 39sylan2b 275 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( C R D ) )
4140mpteq2dva 3875 . 2  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( C R D ) ) )
4231, 16, 413eqtrd 2092 1  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    i^i cin 2944    |-> cmpt 3846   dom cdm 4373    Fn wfn 4925   ` cfv 4930  (class class class)co 5540    oFcof 5738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-of 5740
This theorem is referenced by:  fnofval  5749  off  5752  ofres  5753  offval2  5754  suppssof1  5756  ofco  5757  offveqb  5758
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