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Theorem comfffn 13857
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
)
Assertion
Ref Expression
comfffn  |-  O  Fn  ( ( B  X.  B )  X.  B
)

Proof of Theorem comfffn
Dummy variables  x  y  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffn.o . . 3  |-  O  =  (compf `  C )
2 comfffn.b . . 3  |-  B  =  ( Base `  C
)
3 eqid 2387 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2387 . . 3  |-  (comp `  C )  =  (comp `  C )
51, 2, 3, 4comfffval 13851 . 2  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  C
) y ) ,  f  e.  ( (  Hom  `  C ) `  x )  |->  ( g ( x (comp `  C ) y ) f ) ) )
6 ovex 6045 . . 3  |-  ( ( 2nd `  x ) (  Hom  `  C
) y )  e. 
_V
7 fvex 5682 . . 3  |-  ( (  Hom  `  C ) `  x )  e.  _V
86, 7mpt2ex 6364 . 2  |-  ( g  e.  ( ( 2nd `  x ) (  Hom  `  C ) y ) ,  f  e.  ( (  Hom  `  C
) `  x )  |->  ( g ( x (comp `  C )
y ) f ) )  e.  _V
95, 8fnmpt2i 6359 1  |-  O  Fn  ( ( B  X.  B )  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    X. cxp 4816    Fn wfn 5389   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   2ndc2nd 6287   Basecbs 13396    Hom chom 13467  compcco 13468  compfccomf 13819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-comf 13823
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