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Theorem funco 5149
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
StepHypRef Expression
1 funmo 5129 . . . . 5  |-  ( Fun 
G  ->  E* z  x G z )
2 funmo 5129 . . . . . 6  |-  ( Fun 
F  ->  E* y 
z F y )
32alrimiv 2012 . . . . 5  |-  ( Fun 
F  ->  A. z E* y  z F
y )
4 moexexv 2183 . . . . 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 2012 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  A. x E* y E. z ( x G z  /\  z F y ) )
7 funopab 5145 . . 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 4597 . . 3  |-  ( F  o.  G )  =  { <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }
109funeqi 5133 . 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 1532   E.wex 1537   E*wmo 2115   class class class wbr 3920   {copab 3973    o. ccom 4584   Fun wfun 4586
This theorem is referenced by:  fnco  5209  f1co  5303  curry1  6062  curry2  6065  tposfun  6102  fin23lem30  7852  smobeth  8088  hashkf  11217  domrancur1b  24366  domrancur1c  24368
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-fun 4602
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