ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmco GIF version

Theorem dmco 5017
Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
Assertion
Ref Expression
dmco dom (𝐴𝐵) = (𝐵 “ dom 𝐴)

Proof of Theorem dmco
StepHypRef Expression
1 dfdm4 4701 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 4694 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 4737 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 5016 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 4701 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 4849 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2141 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2142 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1316  ccnv 4508  dom cdm 4509  ran crn 4510  cima 4512  ccom 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  casedm  6939  caseinl  6944  caseinr  6945  djudm  6958
  Copyright terms: Public domain W3C validator