ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmptcos Unicode version

Theorem fmptcos 5505
Description: Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fmptcof.1  |-  ( ph  ->  A. x  e.  A  R  e.  B )
fmptcof.2  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
fmptcof.3  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
Assertion
Ref Expression
fmptcos  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
Distinct variable groups:    x, y, B   
y, R    x, S    x, A
Allowed substitution hints:    ph( x, y)    A( y)    R( x)    S( y)    F( x, y)    G( x, y)

Proof of Theorem fmptcos
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fmptcof.1 . 2  |-  ( ph  ->  A. x  e.  A  R  e.  B )
2 fmptcof.2 . 2  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
3 fmptcof.3 . . 3  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
4 nfcv 2235 . . . 4  |-  F/_ z S
5 nfcsb1v 2977 . . . 4  |-  F/_ y [_ z  /  y ]_ S
6 csbeq1a 2955 . . . 4  |-  ( y  =  z  ->  S  =  [_ z  /  y ]_ S )
74, 5, 6cbvmpt 3955 . . 3  |-  ( y  e.  B  |->  S )  =  ( z  e.  B  |->  [_ z  /  y ]_ S )
83, 7syl6eq 2143 . 2  |-  ( ph  ->  G  =  ( z  e.  B  |->  [_ z  /  y ]_ S
) )
9 csbeq1 2950 . 2  |-  ( z  =  R  ->  [_ z  /  y ]_ S  =  [_ R  /  y ]_ S )
101, 2, 8, 9fmptcof 5504 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445   A.wral 2370   [_csb 2947    |-> cmpt 3921    o. ccom 4471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057
This theorem is referenced by:  fmpt2co  6019
  Copyright terms: Public domain W3C validator