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

Theorem dmco 5181
 Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
Assertion
Ref Expression
dmco

Proof of Theorem dmco
StepHypRef Expression
1 dfdm4 4872 . 2
2 cnvco 4865 . . 3
32rneqi 4905 . 2
4 rnco2 5180 . . 3
5 dfdm4 4872 . . . 4
65imaeq2i 5010 . . 3
74, 6eqtr4i 2306 . 2
81, 3, 73eqtri 2307 1
 Colors of variables: wff set class Syntax hints:   wceq 1623  ccnv 4688   cdm 4689   crn 4690  cima 4692   ccom 4693 This theorem is referenced by:  curry1  6210  curry2  6213  smobeth  8208  hashkf  11339  imasless  13442  xppreima  23211  domrancur1b  25200  domrancur1c  25202 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
 Copyright terms: Public domain W3C validator