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Theorem funco 5450
Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funco  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )

Proof of Theorem funco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmo 5429 . . . . 5  |-  ( Fun 
G  ->  E* z  x G z )
2 funmo 5429 . . . . . 6  |-  ( Fun 
F  ->  E* y 
z F y )
32alrimiv 1638 . . . . 5  |-  ( Fun 
F  ->  A. z E* y  z F
y )
4 moexexv 2324 . . . . 5  |-  ( ( E* z  x G z  /\  A. z E* y  z F
y )  ->  E* y E. z ( x G z  /\  z F y ) )
51, 3, 4syl2anr 465 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  E* y E. z ( x G z  /\  z F y ) )
65alrimiv 1638 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  A. x E* y E. z ( x G z  /\  z F y ) )
7 funopab 5445 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }  <->  A. x E* y E. z ( x G z  /\  z F y ) )
86, 7sylibr 204 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
9 df-co 4846 . . 3  |-  ( F  o.  G )  =  { <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }
109funeqi 5433 . 2  |-  ( Fun  ( F  o.  G
)  <->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
118, 10sylibr 204 1  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547   E*wmo 2255   class class class wbr 4172   {copab 4225    o. ccom 4841   Fun wfun 5407
This theorem is referenced by:  fnco  5512  f1co  5607  curry1  6397  curry2  6400  tposfun  6454  fin23lem30  8178  smobeth  8417  hashkf  11575  xppreima  24012  funresfunco  27856
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-fun 5415
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