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Theorem comfffn 13607
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 2283 . . 3  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2283 . . 3  |-  (comp `  C )  =  (comp `  C )
51, 2, 3, 4comfffval 13601 . 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 5883 . . 3  |-  ( ( 2nd `  x ) (  Hom  `  C
) y )  e. 
_V
7 fvex 5539 . . 3  |-  ( (  Hom  `  C ) `  x )  e.  _V
86, 7mpt2ex 6198 . 2  |-  ( g  e.  ( ( 2nd `  x ) (  Hom  `  C ) y ) ,  f  e.  ( (  Hom  `  C
) `  x )  |->  ( g ( x (comp `  C )
y ) f ) )  e.  _V
95, 8fnmpt2i 6193 1  |-  O  Fn  ( ( B  X.  B )  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220  compfccomf 13569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-comf 13573
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