Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funco Structured version   Unicode version

Theorem funco 5483
 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 5462 . . . . 5
2 funmo 5462 . . . . . 6
32alrimiv 1641 . . . . 5
4 moexexv 2350 . . . . 5
51, 3, 4syl2anr 465 . . . 4
65alrimiv 1641 . . 3
7 funopab 5478 . . 3
86, 7sylibr 204 . 2
9 df-co 4879 . . 3
109funeqi 5466 . 2
118, 10sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550  wmo 2281   class class class wbr 4204  copab 4257   ccom 4874   wfun 5440 This theorem is referenced by:  fnco  5545  f1co  5640  curry1  6430  curry2  6433  tposfun  6487  fin23lem30  8212  smobeth  8451  hashkf  11610  xppreima  24049  funresfunco  27920 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-fun 5448
 Copyright terms: Public domain W3C validator