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Theorem fmpt2co 5779
Description: Composition of two functions. Variation of fmptco 5273 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
fmpt2co.1 
R  C
fmpt2co.2  F  ,  |->  R
fmpt2co.3  G  C  |->  S
fmpt2co.4  R  S  T
Assertion
Ref Expression
fmpt2co  G  o.  F  ,  |->  T
Distinct variable groups:   ,,   ,, C,   ,,   , S,   ,,   , R   , T
Allowed substitution hints:   ()   ()   ()    R(,)    S()    T(,)    F(,,)    G(,,)

Proof of Theorem fmpt2co
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmpt2co.1 . . . . . 6 
R  C
21ralrimivva 2395 . . . . 5  R  C
3 eqid 2037 . . . . . 6  ,  |->  R  ,  |->  R
43fmpt2 5769 . . . . 5  R  C  ,  |->  R :  X.  --> C
52, 4sylib 127 . . . 4  ,  |->  R :  X.  --> C
6 nfcv 2175 . . . . . . 7  F/_ R
7 nfcv 2175 . . . . . . 7  F/_ R
8 nfcv 2175 . . . . . . . 8  F/_
9 nfcsb1v 2876 . . . . . . . 8  F/_ [_  ]_ R
108, 9nfcsb 2878 . . . . . . 7  F/_ [_  ]_ [_  ]_ R
11 nfcsb1v 2876 . . . . . . 7  F/_ [_  ]_ [_  ]_ R
12 csbeq1a 2854 . . . . . . . 8  R  [_  ]_ R
13 csbeq1a 2854 . . . . . . . 8  [_  ]_ R 
[_  ]_ [_  ]_ R
1412, 13sylan9eq 2089 . . . . . . 7  R  [_  ]_ [_  ]_ R
156, 7, 10, 11, 14cbvmpt2 5525 . . . . . 6  ,  |->  R  ,  |->  [_  ]_ [_  ]_ R
16 vex 2554 . . . . . . . . . 10 
_V
17 vex 2554 . . . . . . . . . 10 
_V
1816, 17op2ndd 5718 . . . . . . . . 9  <. , 
>.  2nd `
1918csbeq1d 2852 . . . . . . . 8  <. , 
>.  [_ 2nd `  ]_ [_ 1st `  ]_ R  [_  ]_ [_ 1st `  ]_ R
2016, 17op1std 5717 . . . . . . . . . 10  <. , 
>.  1st `
2120csbeq1d 2852 . . . . . . . . 9  <. , 
>.  [_ 1st `  ]_ R  [_  ]_ R
2221csbeq2dv 2869 . . . . . . . 8  <. , 
>.  [_  ]_ [_ 1st `  ]_ R  [_  ]_ [_  ]_ R
2319, 22eqtrd 2069 . . . . . . 7  <. , 
>.  [_ 2nd `  ]_ [_ 1st `  ]_ R  [_  ]_ [_  ]_ R
2423mpt2mpt 5538 . . . . . 6  X.  |->  [_ 2nd `  ]_ [_ 1st `  ]_ R  ,  |->  [_  ]_ [_  ]_ R
2515, 24eqtr4i 2060 . . . . 5  ,  |->  R  X.  |->  [_ 2nd `  ]_ [_ 1st `  ]_ R
2625fmpt 5262 . . . 4  X.  [_ 2nd `  ]_ [_ 1st `  ]_ R  C  ,  |->  R :  X.  --> C
275, 26sylibr 137 . . 3  X. 
[_ 2nd `  ]_ [_ 1st `  ]_ R  C
28 fmpt2co.2 . . . 4  F  ,  |->  R
2928, 25syl6eq 2085 . . 3  F  X.  |->  [_ 2nd `  ]_ [_ 1st `  ]_ R
30 fmpt2co.3 . . 3  G  C  |->  S
3127, 29, 30fmptcos 5275 . 2  G  o.  F  X.  |->  [_ [_ 2nd `  ]_ [_ 1st `  ]_ R  ]_ S
3223csbeq1d 2852 . . . . 5  <. , 
>.  [_ [_ 2nd `  ]_ [_ 1st `  ]_ R  ]_ S  [_ [_  ]_ [_  ]_ R  ]_ S
3332mpt2mpt 5538 . . . 4  X.  |->  [_ [_ 2nd `  ]_ [_ 1st `  ]_ R  ]_ S  ,  |->  [_ [_  ]_ [_  ]_ R  ]_ S
34 nfcv 2175 . . . . 5  F/_ [_ R  ]_ S
35 nfcv 2175 . . . . 5  F/_ [_ R  ]_ S
36 nfcv 2175 . . . . . 6  F/_ S
3710, 36nfcsb 2878 . . . . 5  F/_ [_
[_  ]_ [_  ]_ R  ]_ S
38 nfcv 2175 . . . . . 6  F/_ S
3911, 38nfcsb 2878 . . . . 5  F/_ [_ [_  ]_ [_  ]_ R  ]_ S
4014csbeq1d 2852 . . . . 5  [_ R  ]_ S  [_ [_  ]_ [_  ]_ R  ]_ S
4134, 35, 37, 39, 40cbvmpt2 5525 . . . 4  ,  |->  [_ R  ]_ S  ,  |->  [_ [_  ]_ [_  ]_ R  ]_ S
4233, 41eqtr4i 2060 . . 3  X.  |->  [_ [_ 2nd `  ]_ [_ 1st `  ]_ R  ]_ S  ,  |->  [_ R  ]_ S
4313impb 1099 . . . . 5  R  C
44 nfcvd 2176 . . . . . 6  R  C  F/_ T
45 fmpt2co.4 . . . . . 6  R  S  T
4644, 45csbiegf 2884 . . . . 5  R  C  [_ R  ]_ S  T
4743, 46syl 14 . . . 4  [_ R  ]_ S  T
4847mpt2eq3dva 5511 . . 3  ,  |->  [_ R  ]_ S  ,  |->  T
4942, 48syl5eq 2081 . 2  X. 
|->  [_ [_ 2nd `  ]_ [_ 1st `  ]_ R  ]_ S  ,  |->  T
5031, 49eqtrd 2069 1  G  o.  F  ,  |->  T
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  wral 2300   [_csb 2846   <.cop 3370    |-> cmpt 3809    X. cxp 4286    o. ccom 4292   -->wf 4841   ` cfv 4845    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708
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-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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  oprabco  5780
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