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Theorem comffn 14663
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 2443 . . 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 6135 . . 3  |-  ( g ( <. X ,  Y >. (comp `  C ) Z ) f )  e.  _V
31, 2fnmpt2i 6662 . 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 2443 . . . 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 14657 . . 3  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >. (comp `  C ) Z ) f ) ) )
1211fneq1d 5520 . 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 1369    e. wcel 1756   <.cop 3902    X. cxp 4857    Fn wfn 5432   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   Basecbs 14193   Hom chom 14268  compcco 14269  compfccomf 14624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-comf 14628
This theorem is referenced by: (None)
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