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Theorem comffn 13624
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 2296 . . 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 5899 . . 3  |-  ( g ( <. X ,  Y >. (comp `  C ) Z ) f )  e.  _V
31, 2fnmpt2i 6209 . 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 2296 . . . 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 13618 . . 3  |-  ( ph  ->  ( <. X ,  Y >. O Z )  =  ( g  e.  ( Y H Z ) ,  f  e.  ( X H Y ) 
|->  ( g ( <. X ,  Y >. (comp `  C ) Z ) f ) ) )
1211fneq1d 5351 . 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 224 1  |-  ( ph  ->  ( <. X ,  Y >. O Z )  Fn  ( ( Y H Z )  X.  ( X H Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164    Hom chom 13235  compcco 13236  compfccomf 13585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-comf 13589
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