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Theorem funco 5631
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

Proof of Theorem funco
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmo 5609 . . . . 5
2 funmo 5609 . . . . . 6
32alrimiv 1719 . . . . 5
4 moexexv 2364 . . . . 5
51, 3, 4syl2anr 478 . . . 4
65alrimiv 1719 . . 3
7 funopab 5626 . . 3
86, 7sylibr 212 . 2
9 df-co 5013 . . 3
109funeqi 5613 . 2
118, 10sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  E.wex 1612  E*wmo 2283   class class class wbr 4452  {copab 4509  o.ccom 5008  Funwfun 5587
This theorem is referenced by:  fnco  5694  f1co  5795  curry1  6892  curry2  6895  tposfun  6990  fsuppco  7881  fsuppco2  7882  fsuppcor  7883  fin23lem30  8743  smobeth  8982  hashkf  12407  xppreima  27487  funresfunco  32210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-fun 5595
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