MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  comffn Structured version   Unicode version

Theorem comffn 15111
Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffn.o  |-  O  =  (compf `  C )
comfffn.b  |-  B  =  ( Base `  C
)
comffn.h  |-  H  =  ( Hom  `  C
)
comffn.x  |-  ( ph  ->  X  e.  B )
comffn.y  |-  ( ph  ->  Y  e.  B )
comffn.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
comffn  |-  ( ph  ->  ( <. X ,  Y >. O Z )  Fn  ( ( Y H Z )  X.  ( X H Y ) ) )

Proof of Theorem comffn
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2382 . . 3  |-  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >. (comp `  C ) Z ) f ) )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >. (comp `  C ) Z ) f ) )
2 ovex 6224 . . 3  |-  ( g ( <. X ,  Y >. (comp `  C ) Z ) f )  e.  _V
31, 2fnmpt2i 6768 . 2  |-  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g ( <. X ,  Y >. (comp `  C ) Z ) f ) )  Fn  ( ( Y H Z )  X.  ( X H Y ) )
4 comfffn.o . . . 4  |-  O  =  (compf `  C )
5 comfffn.b . . . 4  |-  B  =  ( Base `  C
)
6 comffn.h . . . 4  |-  H  =  ( Hom  `  C
)
7 eqid 2382 . . . 4  |-  (comp `  C )  =  (comp `  C )
8 comffn.x . . . 4  |-  ( ph  ->  X  e.  B )
9 comffn.y . . . 4  |-  ( ph  ->  Y  e.  B )
10 comffn.z . . . 4  |-  ( ph  ->  Z  e.  B )
114, 5, 6, 7, 8, 9, 10comffval 15105 . . 3  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >. (comp `  C ) Z ) f ) ) )
1211fneq1d 5579 . 2  |-  ( ph  ->  ( ( <. X ,  Y >. O Z )  Fn  ( ( Y H Z )  X.  ( X H Y ) )  <->  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y )  |->  ( g (
<. X ,  Y >. (comp `  C ) Z ) f ) )  Fn  ( ( Y H Z )  X.  ( X H Y ) ) ) )
133, 12mpbiri 233 1  |-  ( ph  ->  ( <. X ,  Y >. O Z )  Fn  ( ( Y H Z )  X.  ( X H Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   <.cop 3950    X. cxp 4911    Fn wfn 5491   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   Basecbs 14634   Hom chom 14713  compcco 14714  compfccomf 15074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-comf 15078
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator