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Theorem funco 5374
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 5353 . . . . 5  |-  ( Fun 
G  ->  E* z  x G z )
2 funmo 5353 . . . . . 6  |-  ( Fun 
F  ->  E* y 
z F y )
32alrimiv 1631 . . . . 5  |-  ( Fun 
F  ->  A. z E* y  z F
y )
4 moexexv 2279 . . . . 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 464 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  E* y E. z ( x G z  /\  z F y ) )
65alrimiv 1631 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  A. x E* y E. z ( x G z  /\  z F y ) )
7 funopab 5369 . . 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 203 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
9 df-co 4780 . . 3  |-  ( F  o.  G )  =  { <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }
109funeqi 5357 . 2  |-  ( Fun  ( F  o.  G
)  <->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
118, 10sylibr 203 1  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1540   E.wex 1541   E*wmo 2210   class class class wbr 4104   {copab 4157    o. ccom 4775   Fun wfun 5331
This theorem is referenced by:  fnco  5434  f1co  5529  curry1  6297  curry2  6300  tposfun  6337  fin23lem30  8058  smobeth  8298  hashkf  11432  xppreima  23262  funresfunco  27313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-fun 5339
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