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Theorem offval 5661
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1  F  Fn
offval.2  G  Fn
offval.3  V
offval.4  W
offval.5  i^i  S
offval.6  F `  C
offval.7  G `  D
Assertion
Ref Expression
offval  F  o F R G  S  |->  C R D
Distinct variable groups:   ,   , F   , G   ,   , S   , R
Allowed substitution hints:   ()    C()    D()    V()    W()

Proof of Theorem offval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4  F  Fn
2 offval.3 . . . 4  V
3 fnex 5326 . . . 4  F  Fn  V  F  _V
41, 2, 3syl2anc 391 . . 3  F  _V
5 offval.2 . . . 4  G  Fn
6 offval.4 . . . 4  W
7 fnex 5326 . . . 4  G  Fn  W  G  _V
85, 6, 7syl2anc 391 . . 3  G  _V
9 fndm 4941 . . . . . . . 8  F  Fn  dom  F
101, 9syl 14 . . . . . . 7  dom  F
11 fndm 4941 . . . . . . . 8  G  Fn  dom  G
125, 11syl 14 . . . . . . 7  dom  G
1310, 12ineq12d 3133 . . . . . 6  dom  F  i^i  dom 
G  i^i
14 offval.5 . . . . . 6  i^i  S
1513, 14syl6eq 2085 . . . . 5  dom  F  i^i  dom 
G  S
1615mpteq1d 3833 . . . 4  dom  F  i^i  dom  G  |->  F `
 R G `  S  |->  F `  R G `
17 inex1g 3884 . . . . . 6  V  i^i 
_V
1814, 17syl5eqelr 2122 . . . . 5  V  S  _V
19 mptexg 5329 . . . . 5  S  _V  S  |->  F `  R G ` 
_V
202, 18, 193syl 17 . . . 4  S  |->  F `  R G `
 _V
2116, 20eqeltrd 2111 . . 3  dom  F  i^i  dom  G  |->  F `
 R G `  _V
22 dmeq 4478 . . . . . 6  F  dom  dom  F
23 dmeq 4478 . . . . . 6  G  dom  dom  G
2422, 23ineqan12d 3134 . . . . 5  F  G  dom  i^i 
dom  dom  F  i^i  dom 
G
25 fveq1 5120 . . . . . 6  F  `  F `
26 fveq1 5120 . . . . . 6  G  `  G `
2725, 26oveqan12d 5474 . . . . 5  F  G  `  R `
 F `
 R G `
2824, 27mpteq12dv 3830 . . . 4  F  G  dom  i^i  dom  |->  `  R `  dom  F  i^i  dom  G  |->  F `
 R G `
29 df-of 5654 . . . 4  o F R  _V ,  _V  |->  dom  i^i  dom  |->  `  R `
3028, 29ovmpt2ga 5572 . . 3  F  _V  G  _V  dom  F  i^i  dom  G  |->  F `  R G `
 _V  F  o F R G  dom  F  i^i  dom  G  |->  F `
 R G `
314, 8, 21, 30syl3anc 1134 . 2  F  o F R G  dom  F  i^i  dom  G  |->  F `
 R G `
3214eleq2i 2101 . . . . 5  i^i  S
33 elin 3120 . . . . 5  i^i
3432, 33bitr3i 175 . . . 4  S
35 offval.6 . . . . . 6  F `  C
3635adantrr 448 . . . . 5  F `  C
37 offval.7 . . . . . 6  G `  D
3837adantrl 447 . . . . 5  G `  D
3936, 38oveq12d 5473 . . . 4  F `  R G `  C R D
4034, 39sylan2b 271 . . 3  S  F `
 R G `  C R D
4140mpteq2dva 3838 . 2  S  |->  F `  R G `
 S  |->  C R D
4231, 16, 413eqtrd 2073 1  F  o F R G  S  |->  C R D
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551    i^i cin 2910    |-> cmpt 3809   dom cdm 4288    Fn wfn 4840   ` cfv 4845  (class class class)co 5455    o Fcof 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654
This theorem is referenced by:  fnofval  5663  off  5666  ofres  5667  offval2  5668  suppssof1  5670  ofco  5671  offveqb  5672
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