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Theorem funco 5259
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 ) )
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem funco
StepHypRef Expression
1 funmo 5239 . . . . 5  |-  ( Fun 
G  ->  E* z  x G z )
2 funmo 5239 . . . . . 6  |-  ( Fun 
F  ->  E* y 
z F y )
32alrimiv 1619 . . . . 5  |-  ( Fun 
F  ->  A. z E* y  z F
y )
4 moexexv 2216 . . . . 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 466 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  E* y E. z ( x G z  /\  z F y ) )
65alrimiv 1619 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  A. x E* y E. z ( x G z  /\  z F y ) )
7 funopab 5255 . . 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 205 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
9 df-co 4699 . . 3  |-  ( F  o.  G )  =  { <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }
109funeqi 5243 . 2  |-  ( Fun  ( F  o.  G
)  <->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
118, 10sylibr 205 1  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1529   E.wex 1530   E*wmo 2147   class class class wbr 4026   {copab 4079    o. ccom 4694   Fun wfun 5217
This theorem is referenced by:  fnco  5319  f1co  5413  curry1  6173  curry2  6176  tposfun  6213  fin23lem30  7965  smobeth  8205  hashkf  11335  domrancur1b  24601  domrancur1c  24603  funresfunco  27369
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pr 4215
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-op 3652  df-br 4027  df-opab 4081  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-fun 5225
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