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Theorem offval2 5668
Description: The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval2.1  V
offval2.2  W
offval2.3  C  X
offval2.4  F  |->
offval2.5  G  |->  C
Assertion
Ref Expression
offval2  F  o F R G  |->  R C
Distinct variable groups:   ,   ,   , R
Allowed substitution hints:   ()    C()    F()    G()    V()    W()    X()

Proof of Theorem offval2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 offval2.2 . . . . . 6  W
21ralrimiva 2386 . . . . 5  W
3 eqid 2037 . . . . . 6  |->  |->
43fnmpt 4968 . . . . 5  W  |->  Fn
52, 4syl 14 . . . 4  |->  Fn
6 offval2.4 . . . . 5  F  |->
76fneq1d 4932 . . . 4  F  Fn  |->  Fn
85, 7mpbird 156 . . 3  F  Fn
9 offval2.3 . . . . . 6  C  X
109ralrimiva 2386 . . . . 5  C  X
11 eqid 2037 . . . . . 6  |->  C  |->  C
1211fnmpt 4968 . . . . 5  C  X  |->  C  Fn
1310, 12syl 14 . . . 4  |->  C  Fn
14 offval2.5 . . . . 5  G  |->  C
1514fneq1d 4932 . . . 4  G  Fn  |->  C  Fn
1613, 15mpbird 156 . . 3  G  Fn
17 offval2.1 . . 3  V
18 inidm 3140 . . 3  i^i
196adantr 261 . . . 4  F  |->
2019fveq1d 5123 . . 3  F `  |->  `
2114adantr 261 . . . 4  G  |->  C
2221fveq1d 5123 . . 3  G `  |->  C `
238, 16, 17, 17, 18, 20, 22offval 5661 . 2  F  o F R G  |->  |->  `  R  |->  C `
24 nffvmpt1 5129 . . . . 5  F/_  |->  `
25 nfcv 2175 . . . . 5  F/_ R
26 nffvmpt1 5129 . . . . 5  F/_  |->  C `
2724, 25, 26nfov 5478 . . . 4  F/_  |->  `  R  |->  C `
28 nfcv 2175 . . . 4  F/_  |->  `  R  |->  C `
29 fveq2 5121 . . . . 5  |->  `  |->  `
30 fveq2 5121 . . . . 5  |->  C `  |->  C `
3129, 30oveq12d 5473 . . . 4  |->  `  R  |->  C `
 |->  `  R  |->  C `
3227, 28, 31cbvmpt 3842 . . 3  |->  |->  `  R  |->  C `
 |->  |->  `  R  |->  C `
33 simpr 103 . . . . . 6
343fvmpt2 5197 . . . . . 6  W  |->  `
3533, 1, 34syl2anc 391 . . . . 5  |->  `
3611fvmpt2 5197 . . . . . 6  C  X  |->  C `  C
3733, 9, 36syl2anc 391 . . . . 5  |->  C `
 C
3835, 37oveq12d 5473 . . . 4  |->  `  R  |->  C `
 R C
3938mpteq2dva 3838 . . 3  |->  |->  `  R  |->  C `  |->  R C
4032, 39syl5eq 2081 . 2  |->  |->  `  R  |->  C `  |->  R C
4123, 40eqtrd 2069 1  F  o F R G  |->  R C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390  wral 2300    |-> cmpt 3809    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:  ofc12  5673  caofinvl  5675  caofcom  5676
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